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PROHITECH EARTHQUAKE PROTECTION OF HISTORICAL BUILDINGS BY REVERSIBLE

MIXED TECHNOLOGIES

WORK PACKAGE 9 DEVELOPMENT OF CALCULATION MODELS

Draft

WP Leader: Dan Dubina, Politehnica University of Timisoara, Romania (ROPUT)

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Main Deliverables D III - Reversible mixed technologies for seismic protection: set-up of calculation methods

Deliverables: D10 Set-up of analytical models for special materials and special devices for the seismic structural control; D11 Development of simplified models for the global seismic analysis of historical constructions.

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List of content

LIST OF CONTENT......................................................................................................................................................................3 INTRODUCTION..........................................................................................................................................................................8 INTEGRATED SUMMARY OF CALCULATION MODELS..............................................................................................10 D10 SET-UP OF ANALYTICAL MODELS FOR SPECIAL MATERIALS AND SPECIAL DEVICES FOR THE SEISMIC STRUCTURAL CONTROL.....................................................................................................................................18 1. MODELS AND PERFORMANCE CRITERIA FOR STRUCTURAL ELEMENTS OF DIFFERENT MATERIAL ..................................................................................................................................................................................18

1.1. MASONRY ELEMENTS CLAY BRICK ..........................................................................................................................18 1.1.1. Introduction............................................................................................................................................................18 1.1.2. General problems of the material and element behavior......................................................................................18 1.1.3. Models for masonry component materials (Material model) ...............................................................................21 1.1.4. Models for masonry as a composite material (Element model walls)...............................................................21 1.1.5. Analysis types and performance criteria...............................................................................................................36 1.1.6. Design assisted by testing:.....................................................................................................................................36 1.1.7. References: .............................................................................................................................................................36

1.2. MARBLE AND LIMESTONE............................................................................................................................................39 1.2.1. Introduction............................................................................................................................................................39 1.2.2. Material properties ................................................................................................................................................39 1.2.3. Linear analysis parameters - Youngs modulus and Poissons ratio ...................................................................40 1.2.4. Tensile strength ......................................................................................................................................................40 1.2.5. Fracture toughness ................................................................................................................................................42 1.2.6. Reference:...............................................................................................................................................................44

1.3. CONCRETE / REINFORCED CONCRETE..........................................................................................................................44 1.3.1. Materials ................................................................................................................................................................44 1.3.2. Modelling of elements ............................................................................................................................................45 1.3.3. Beam effective width ..............................................................................................................................................48 1.3.4. Beam-column joints................................................................................................................................................51 1.3.5. Shear resistance of members .................................................................................................................................56 1.3.6. Anchorage failure ..................................................................................................................................................58 1.3.7. Rotation capacity of elements ................................................................................................................................59 1.3.8. References ..............................................................................................................................................................59

1.4. IRON ELEMENTS ...........................................................................................................................................................60 1.4.1. Introduction............................................................................................................................................................60 1.4.2. Material model .......................................................................................................................................................61 1.4.3. Element model ........................................................................................................................................................61 1.4.4. Reference:...............................................................................................................................................................64

1.5. TIMBER ELEMENTS ......................................................................................................................................................65 1.5.1. Introduction............................................................................................................................................................65 1.5.2. Models for material................................................................................................................................................65 1.5.3. Models for Elements...............................................................................................................................................69 1.5.4. Connections............................................................................................................................................................69 1.5.5. System model ..........................................................................................................................................................70 1.5.6. Analysis type...........................................................................................................................................................70 1.5.7. Reference:...............................................................................................................................................................70

2. MODELS AND PERFORMANCE CRITERIA FOR STRUCTURAL ELEMENTS OF DIFFERENT MATERIAL ..................................................................................................................................................................................72

2.1. RIVETED CONNECTION.................................................................................................................................................72 2.1.1. Introduction............................................................................................................................................................72 2.1.2. Hand made calculation models .............................................................................................................................72 2.1.3. Numerical vs. Hand made calculation ..................................................................................................................77 2.1.4. Experimental vs. predicted shear strength ............................................................................................................77 2.1.5. References ..............................................................................................................................................................78

2.2. ARCHITRAVE CONNECTION..........................................................................................................................................78

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2.2.1. Introduction............................................................................................................................................................78 2.2.2. Basic concept .........................................................................................................................................................78 2.2.3. Architrave connection tests with main results.......................................................................................................80 2.2.4. Numerical simulation with applied fracture criteria ............................................................................................84 2.2.5. Large scale modelling of ancient temples with columns and architraves ............................................................93 2.2.6. References: .............................................................................................................................................................94

2.3. ANCHORS IN MARBLE ..................................................................................................................................................94 2.3.1. Introduction............................................................................................................................................................94 2.3.2. Basic concept .........................................................................................................................................................95 2.3.3. Summary experimental results...............................................................................................................................96 2.3.4. A design criterion based on the experimental results ...........................................................................................98 2.3.5. Practical implementation of the design criterion................................................................................................101 2.3.6. Conclusions ..........................................................................................................................................................102 2.3.7. References: ...........................................................................................................................................................102

2.4. POST-INSTALLED ANCHORS IN CONCRETE.................................................................................................................103 2.4.1. Introduction..........................................................................................................................................................103 2.4.2. Design of post-installed anchors .........................................................................................................................103 2.4.3. Types and methods of post-installed anchors......................................................................................................104 2.4.4. Material, shape and dimensions of post-installed anchors.................................................................................106 2.4.5. Design strength ....................................................................................................................................................108 2.4.6. Applying post-installed anchors in low-strength concrete .................................................................................111 2.4.7. Structural design in the case of shear RC wall ...................................................................................................113 2.4.8. References: ...........................................................................................................................................................117

2.5. PURE ALUMINIUM SHEAR PANELS..............................................................................................................................117 2.5.1. Introduction..........................................................................................................................................................117 2.5.2. Description of the device .....................................................................................................................................117 2.5.3. Material model .....................................................................................................................................................119 2.5.4. System model ........................................................................................................................................................120 2.5.5. Analysis procedure...............................................................................................................................................124 2.5.6. References ............................................................................................................................................................124

2.6. MAGNETORHEOLOGICAL DEVICES ............................................................................................................................125 2.6.1. Description of the device .....................................................................................................................................125 2.6.2. Device models ......................................................................................................................................................126 2.6.3. References: ...........................................................................................................................................................130

2.7. STEEL BUCKLING RESTRAINED BRACES.....................................................................................................................136 2.7.1. Introduction..........................................................................................................................................................136 2.7.2. Description of the device/technique ....................................................................................................................137 2.7.3. Material model .....................................................................................................................................................139 2.7.4. Element model ......................................................................................................................................................139 2.7.5. Connections..........................................................................................................................................................146 2.7.6. System model ........................................................................................................................................................147 2.7.7. Analysis types .......................................................................................................................................................148 2.7.8. Performance criteria............................................................................................................................................148 2.7.9. References ............................................................................................................................................................150

2.8. EBF ECCENTRIC BRACED FRAMES..........................................................................................................................152 2.8.1. Description of EB technique ................................................................................................................................152 2.8.2. Analytical methods for the design of EB .............................................................................................................155 2.8.3. References: ...........................................................................................................................................................160

2.9. METAL SHEAR PANEL ................................................................................................................................................161 2.9.1. Introduction..........................................................................................................................................................161 2.9.2. Description of the device/technique ....................................................................................................................161 2.9.3. Interpreting models ..............................................................................................................................................162 2.9.4. References: ...........................................................................................................................................................164

2.10. FRP FIBER REINFORCED POLIMERS ........................................................................................................................165 2.10.1. Introduction .....................................................................................................................................................165 2.10.2. Description of the FPR device/technique .......................................................................................................165 2.10.3. Analytical methods for the design of FRP......................................................................................................166 2.10.4. Masonry reinforcement ...................................................................................................................................172 2.10.5. Connections .....................................................................................................................................................172 2.10.6. Pefrormance criteria.......................................................................................................................................174

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2.10.7. References: ......................................................................................................................................................174 2.11. PIN INERD ...............................................................................................................................................................176

2.11.1. Introduction .....................................................................................................................................................176 2.11.2. Description of the device/technique................................................................................................................176 2.11.3. Behavior of INERD connections.....................................................................................................................177 2.11.4. Design reules for the INERD connection .......................................................................................................178 2.11.5. Capacity design criteria..................................................................................................................................180 2.11.6. System model ...................................................................................................................................................180 2.11.7. Analysis types ..................................................................................................................................................180 2.11.8. Performance criteria.......................................................................................................................................180 2.11.9. References: ......................................................................................................................................................181

3. MODELS AND PERFORMANCE CRITERIA FOR SUB-SYSTEMS ....................................................................182 3.1. MASONRY WALLS STRENGTHENING WITH METAL BASED TECHNIQUES ....................................................................182

3.1.1. Introduction..........................................................................................................................................................182 3.1.2. Basic concept .......................................................................................................................................................182 3.1.3. Design assisted by testing (see Chapter 1)..........................................................................................................183 3.1.4. Analytical calibration ..........................................................................................................................................184 3.1.5. Experimental calibration .....................................................................................................................................185 3.1.6. Numerical approaches .........................................................................................................................................187 3.1.7. Reference..............................................................................................................................................................188

3.2. CONFINED MASONRY .................................................................................................................................................189 3.2.1. Introduction..........................................................................................................................................................189 3.2.2. General.................................................................................................................................................................189 3.2.3. In-plane stiffness evaluation of masonry-infill walls ..........................................................................................191 3.2.4. In-plane strength evaluation of masonry-infill walls ..........................................................................................193 3.2.5. Infill with openings...............................................................................................................................................196 3.2.6. Application to the case study ...............................................................................................................................198 3.2.7. References ............................................................................................................................................................204

3.3. MASONRY WALLS STRENGTHENING WITH FRP COMPOSITES....................................................................................206 3.3.1. Introduction..........................................................................................................................................................206 3.3.2. Generalities ..........................................................................................................................................................206 3.3.3. In-plane shear capacity evaluation of unreinforced masonry walls strengthened with FRP composites .........207 3.3.4. In-plane strength evaluation of masonry-infill walls strengthened with FRP systems ......................................209 3.3.5. References ............................................................................................................................................................209

3.4. REINFORCED CONCRETE STRUCTURES RETROFFITED WITH STEEL JACKETING .........................................................210 3.4.1. Introduction..........................................................................................................................................................211 3.4.2. Example of reinforcement calculation to increase ductility of a column ...........................................................211 3.4.3. Example of reinforcement calculation to increase column resistance to axial force.........................................214 3.4.4. References ............................................................................................................................................................216

3.5. TIMBER COMPOSITE FLOOR........................................................................................................................................216 3.5.1. Introduction..........................................................................................................................................................216 3.5.2. Description of Device / Technique ......................................................................................................................216 3.5.3. Material Model.....................................................................................................................................................216 3.5.4. Element Model .....................................................................................................................................................218 3.5.5. Connections..........................................................................................................................................................221 3.5.6. System Model .......................................................................................................................................................221 Analytical Method Considering Slip ..................................................................................................................................222 Slip Moduli..........................................................................................................................................................................226 3.5.7. Analysis Types......................................................................................................................................................227 3.5.8. Performance Criteria...........................................................................................................................................227 Effective Bending Stiffness .................................................................................................................................................228 Normal Stresses ..................................................................................................................................................................228 3.5.9. References ............................................................................................................................................................229

3.6. DEVELOPMENT OF DESIGN RULES FOR THE IRON COLUMNS REINFORCED BY FRP ...................................................229 3.6.1. Description of the device/technique ....................................................................................................................230 3.6.2. Material model .....................................................................................................................................................230 FRP material.......................................................................................................................................................................230 3.6.3. Element model ......................................................................................................................................................231 State of the art.....................................................................................................................................................................231

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Cross-sectional resistance in compression ........................................................................................................................232 Members under axial compression ....................................................................................................................................232 Validation of the model with numerical simulations .........................................................................................................235 3.6.4. Conclusions ..........................................................................................................................................................238 Notations .............................................................................................................................................................................239 3.6.5. References ............................................................................................................................................................240

3.7. REINFORCED CONCRETE FRAMES RETROFITED WITH ECCENTRIC BRACES ...............................................................242 3.7.1. Introduction..........................................................................................................................................................242 3.7.2. Connections..........................................................................................................................................................242 3.7.3. Performance criteria............................................................................................................................................244 3.7.4. References ............................................................................................................................................................245

3.8. REINFORCED CONCRETE STRUCTURES RETROFITTED WITH METAL SHEAR PANEL ..................................................246 3.8.1. Introduction..........................................................................................................................................................246 3.8.2. Retrofitting design method...................................................................................................................................246 The general approach.........................................................................................................................................................246 3.8.3. The study case ......................................................................................................................................................248 3.8.4. Application methodology .....................................................................................................................................248 3.8.5. The behavior of the retrofitted structures............................................................................................................249 3.8.6. References ............................................................................................................................................................251

D11 DEVELOPMENT OF SIMPLIFIED MODELS FOR THE GLOBAL SEISMIC ANALYSIS OF HISTORICAL CONSTRUCTIONS ...................................................................................................................................................................253 4. MODELS FOR GLOBAL ANALYSIS..........................................................................................................................253

4.1. ANALYSIS METHODS..................................................................................................................................................253 4.1.1. Global analysis and modeling requirements.......................................................................................................253 4.1.2. Choice of analysis procedure ..............................................................................................................................259 4.1.3. Choice of the intervention technique ...................................................................................................................261 4.1.4. PBE Methodologies and examples ......................................................................................................................263 4.1.5. Vulnerability Analysis ..........................................................................................................................................266 4.1.6. Reference..............................................................................................................................................................269

4.2. OVERVIEW OF COLLAPSE MODES AND EVALUATION OF BEARING CAPACITY............................................................272 4.2.1. Introduction..........................................................................................................................................................272 4.2.2. Generalities ..........................................................................................................................................................272 4.2.3. Calculation models of masonry buildings for seismic design.............................................................................273 Strategies for masonry building modeling .........................................................................................................................273 Performance-based design of masonry buildings..............................................................................................................273 Computations on masonry structures.................................................................................................................................275 Modeling masonry building by rigid blocks ......................................................................................................................278 4.2.4. Ultimate limit states for masonry elements .........................................................................................................280 Masonry walls.....................................................................................................................................................................280 (i) For in-plane behaviour..................................................................................................................................................280 (ii) The case of out-of-plane ...............................................................................................................................................284 4.2.5. Collapse mechanisms for masonry floors, arches, vaults and domes ................................................................288 4.2.6. Ultimate limit state for masonry structures.........................................................................................................291 Collapse mechanisms for buildings....................................................................................................................................291 4.2.7. Collapse mechanisms for individual buildings ...................................................................................................291 4.2.8. Collapse mechanisms for complex buildings ......................................................................................................294 Romanesque churches ........................................................................................................................................................298 Gothic churches ..................................................................................................................................................................298 Renaissance churches.........................................................................................................................................................299 Byzantine churches .............................................................................................................................................................300 4.2.9. Romanesque churches..........................................................................................................................................301 4.2.10. Gothic churches...............................................................................................................................................306 4.2.11. Renaissance churches .....................................................................................................................................308 4.2.12. Byzantine churches..........................................................................................................................................308 4.2.13. Retrofitting of masonry buildings using the collapse mechanisms................................................................309 4.2.14. Conclusions .....................................................................................................................................................311 4.2.15. References .......................................................................................................................................................312

CONCLUDING REMARKS.....................................................................................................................................................315

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ANNEXES ...................................................................................................................................................................................316 LIST OF CONTRIBUTORS AND DATA SHEETS: ............................................................................................................................316

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INTRODUCTION

The activity in WP 9 is belonging to the main research part of PROHITECH Project (Part R3- Experimental and numerical research) aiming to produce the main deliverables D III - Reversible mixed technologies for seismic protection: set-up of calculation methods. WP 9 had the mission to provide practical calculation models (or tools) for structural systems and or devices applied or developed and studied in the parallel Working Packages, WP 7 - Experimental analysis and WP 8 - Numerical analysis. The WP9 specific deliverables are:

D10 Set-up of analytical models for special materials and special devices for the seismic structural control; D11 Development of simplified models for the global seismic analysis of historical constructions.

These deliverables are connected to the Objective no 7 of the project (see B.1.1 in Description of scientific/technological objectives and work-plan) aiming to allow engineers to use simple and reliable tools for analyzing the behaviour of constructions provided with advanced systems for seismic protection, as well as for detailing up-grading interventions.

The calculation models, procedures and tools provided by WP 9 are intended to form the basis of the design guidelines to be developed in WP12 Development of design guidelines.

The following partners have been assigned to contribute to the achievement of tasks related that objective and deliverables D 10 and D11:

1. UNINA (PTN. N 1) F.M. MAZZOLANI 2. B (PTN. N 2) J-P. JASPART 3. MK (PTN. N. 3) K. GRAMATIKOV (corresponding) 4. NA ARC (PTN. N 5) R. LANDOLFO 5. ROPUT (PTN. N 7) D. DUBINA 6. ROTUB (PTN. N 8) D. LUNGU 7. SL (PTN.N. 9) D. BEG (corresponding) 8. TR (PTN. N 10) G. ALTAY (ASKAR) 9. ISR (PTN. N 11) A.V. RUTENBERG 10. EG (PTN. N 12) M. EL ZAHABI 11. SUN (PTN. N 14) A. MANDARA 12. UNICH (PTN. N 16) G. DE MATTEIS

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Their contributions were materialized in 24 Datasheets on the different strengthening techniques and devices associated with different structural materials and systems. The Datasheet have been produced on the basis of a general template of following contents:

Description of the device/technique

Material model

Element model

Connections

System model

Analysis types

Performance criteria

Despite the non-homogeneity of the topic of datasheets, the report tries to structure the essentials of these datasheets in three sections i.e.

I Models and performance criteria for structural elements of different material

II Models for devices and sub-systems tested in the frame of project

III Models for global analysis

The Integrated Summary Table, which precedes these sections, represents the envelope of whole WP 9 contributions.

The datasheets and their authors are presented in the Annex of this Report.

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INTEGRATED SUMMARY OF CALCULATION MODELS

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INTEGRATED SUMMARY OF CALCULATION MODELS

Material Model Element Model DEVICE MODEL System model Connection Analysis type Performance criteria UNREINFORCED CLAY BRICK MASONRY (principles are common to other masonry types)

Capacity design Linear analysis low dissipative

Ultimate strenght associate with relevant failure mode

Composite material linear behavior brittle fracture Analytical formulas: uniaxial behavior,

biaxial behavior,

shear behavior

Masonry shear walls (1.1) Codes relations associated with failure modes: - Sliding (a) - Flexural (b) - Shear (c)

Physical models Equivalent strut confined masonry

Numerical models: Micro-model (Interface - a, Layer model - b), Macro model (Homogeneous - c)

a b c

Discontinuous modeling of masonry Discrete crack approach Smeared crack approach Interface smeared crack approach Use of joint elements

2D systems identical with elements models -unreinforced shear walls composed by piers and spandrel - Capacity of transversal walls - Capacity of longitudinal walls 3D systems can be decomposed in 2D subassamblies

Nonlinear static analisys displacement based metodologies

(values for a, b, c, d, e FEMA 356) Generalized Component Force-Deformation Relations Modeling

Shear deformation of panel

IO Immediate ocupancy LS Life savety CP Collapse prevention Interstory drifts limits

REINFORCED CLAY BRICK MASONRY Metal sheating (3.1)

M

a

s

o

n

r

y

b

a

s

e

d

s

h

e

a

r

w

a

l

l

s

Masonry material model Aluminiu Steel

Composite masonry-metal shear walls External reinforcing of walls

Masonry wall (150x150x25 cm)

Steel/alluminium plate

Chemical anchors

Metal (aluminiu / steel) shear panel connected with chemical anchors or prestressed ties

2D systems - composite shear walls composed by piers and spandrel 3D systems can be decomposed in 2D subassamblies

Constitutive law - Multi-linear spring

0

5000

10000

15000

0 5 10 15 20 25 30

Displacement (mm)

F

o

r

c

e

(

N

)

Mechanical anchor prestress ties Chemical anchor

Capacity design assisted by tests Capacity design: numerical analysis linear static, nonlinear static

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FRP (2.10; 3.3) FRP f,u

Ef

Composite FRP masonry wallss FRP glued with epoxy resin Analytical models

Bonding structure

Failure modes

Mode 1 - Laminate/sheet end debonding; Mode 2 - Intermediate debonding, caused by flexural cracks; Mode 3 - Debonding caused by diagonal shear cracks; Mode 4 - Debonding caused by irregularities and roughness of concrete surface; Bonding length

Glued with epoxy resin Steel wire mesh SWM (zincoated or stainless steel)

Composite masonry steel wire mesh (similar with FRP technologiy)

SWM with epoxy resin Similar with FRP

UNREINFORCED CLAY BRICK MASONRY: RIGID BLOCS ULTIMATE LIMIT ANALYSIS (4.2) - Material simplificate assumtion - rigid material with not tensile strenght

Modeling masonry elements as rigid blocs - sliding failure mode (a); - shear failure mode (b); - overturning failure mode (c).

d

a) b) c)

Unreinforced elements Assemblages of rigid blocks (composing walls) interacting through joints Ussualy 2D models (3D possible) Proposed colapse mechanism

a) b)

c) d)

e)

Joints pinned conection without friction Joints assumption: - the compression and shear failures at the joints are perfectly plastic; - hinging failure at joint does not consider the effects of local crushing.

Ultimate limits state analysis Kinematic analysis of proposed colapse mechanisms

Minimum value of the determined amplification factors determinated by minuimum total potential energy

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Material Model Element model Device model System model Connection Analysis type Performance criteria UNCONNECTED MARBLE OR LIMESTONE BLOCS STRUCTURE

Assemblages of rigid blocks interacting through joints (similar with )

- Self weight - Friction joint

Ultimate limits state analysis Kinematic analysis of proposed colapse mechanisms

Minimum value of the determined amplification factors

Linear behavior (1.2) Numerical models Failure criteria: Tensile stress Fracture toughness

Columns: compression - Axial - Eccentric (rocking effect) Archittraves (dominant shear effect accompaned by bending) Numerical models based on fracture mechanics

Pinned frames Pinned/simple suported conection Capacity design Linear static

Ultimate limit state material strenght Overall stability

CONNECTED MARBLE OR LIMESTONE BLOCS STRUCTURE Marble connectors (2.3) Capacity design

Linear static low dissipative Ulitimate limit state - Material strenght Ultimate slip in connection (elastic deformation of spring)

Schematic representation of the specimens (m: marble, c: cement mat., b: bar)

Clamped joints P-slip curves - Linear behavior linear spring - Bilinear behavior nonlinear spring

0

8

16

24

0 2 4 6 8System displacement, [mm]

L

o

a

d

,

P

[

k

N

]

0

1

2

3

4

5

6

7

8

S

l

i

p

[

m

m

]

P-slip-

First change of slope

Initiation of slip

Peak load

Slip evolution, Vmax, Vmed

Architrave connection (2.2)

M

a

r

b

l

e

a

n

d

l

i

m

e

s

t

o

n

e

Semi-rigid frames

Force F Fu,fract.

Strain

u,clamp

Fu,clamp

Load-strain responseof clamp neck

Bilinear approximation of clamp response

Failure of spring

Nonlinear static analysis (e.g. nonlinear connector)

Material strenght Predifine slip Failure of the spring: - Fracture of the stone - Failure of the clamp

- 13 -

Material Model Element Model Device model System model Connection Analysis type Performance criteria UNREINFORCED IRON

Linear behavior in tension (1.4) Ramberg-Osgood law in compression

Columns Analytical formula Cross sectional resistance Elastic verification Cross section imperfection

Member Geometric imperfection - crookedness Tension failure Compression failure Member stability Member in axial compression Member in bending (LTB) Member in bending and axial compression

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00

Lambda Bi

N

B

Compression failure in thin side

Traction failure in thick side

Numerical model without FRP

2D and 3D framing global imperfection Trusses

Usualy riveted pinned connection - tying - component method physical model (2.1)

Rigid joints of riveted connection can be obtained

- Capacity design - Linear static - Buckling analysis - Nonlinear analysis posible

Allowable stress Limit state criteria - Strenght - Stifness - Stability

REINFORCED IRON FRP (3.6)

I

r

o

n

f

r

a

m

i

n

g

Iron + FRP Elastic behavior FRP

f,u

Ef

Composite element Stress strain relation

Thick side

Thin side

i,t (i,t) v'eq

i,c (i,c) M

geq

N

Stress Strain

FRP

Iron veq

Mechanical characteristics of a composite cross-section Analytical formula Tension failure Compression failure

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00

Lambda Bi

N

B

without FRP, compression failure in thin side

without FRP, traction failure in thick side

with FRP, compression failure in thin side

with FRP, traction failure in thick side

Numerical model

- 14 -

Material Model Element Model Device model System model Connection Analysis type Performance criteria UNREIFORCED TIMBER FRAMES

Capacity design Linear static low dissipative Linear dynamic - Lateral force method - Linear time-history

Ultimate limit state Material strenght

Linear orthotropic behavior: Tension parallel to the grain Tension perpendicular to the grain Compression parallel to the grain Tension parallel to the grain Bending Shear

Linear elements Analytical formula for beams, columns

2D and 3D Pinned frames Trusses

Pined connections Bilinear or multilinear elasto-plastic spring Gap - sliding

Nonlinear static Nonlinear dynamic

Plastic deformation Interstory drift

REIFORCED TIMBER FRAMES Compsite concrete wood floor (3.5) Linear static low dissipative

Linear dynamic Ultimate limit state Material strenght

T

i

m

b

e

r

f

r

a

m

i

n

g

Elastic model Analytical model considering slip

Slip moduli Rigid diaphragm effect

2D and 3D Pinned frames

Numerical models behaviour curves P-

0,4Fu

Fu

F

0,7Fu

kULSkSLS

Nonlinear static Nonlinear dynamic

Plastic deformation Interstorey drift

Material Model Element Model Device model System model Connection Analysis type Performance criteria UNREINFORCED RC FRAMES

Linear static q factor Linear dynamic

Ultimate limit state Material strenght

R

e

i

n

f

o

r

c

e

d

c

o

n

c

r

e

t

e

f

r

a

m

e

s

Uniaxial behavior (1.3)

Biaxial behavior

Columns and beams 2D or 3D moment resisting franes Rigid and semi-rigid connection (in case of pre-cast concrete)

Nonlinear static Nonlinear dynamic

Plastic deformation (rotation) Interstorey drift

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REINFORCED RC FRAMES Aluminum / Steel Panel (2.5, 2.9, 3.8) Linear static - q factor

Linear dynamic Limit state criteria - Strenght - Rigidity - Stability

Aluminium Stress strain relation

AW 1050A ALUMINIUM ALLOY

f02=115 MPa

fu=69 MPa

f02=20 MPa

0

20

40

60

80

100

120

140

0% 10% 20% 30% 40% 50% 60%

strain

s

t

r

e

s

s

(

M

P

a

)

Not heat treated aluminiumHeat treated aluminium

Stiffned - Unstiffned Classification: compact, semi-compact, slender Analytical formula for strenght and stiffness

Discrete physical models Strip model

Continous phisical models: FEM or FSM (plane stress/strain or shell elements)

2D or 3D moment resisting frames braced with shear panels (strip, FEM or FSM)

Postinstaled anchors: mechanical or chemical - shear strenght - tensile strenght

With or without infill steel frame

Full bay or partial bay type Pinned connection in frame analysis

Nonlinear static Design methodology: Equivalent rigidity and strength

Line of minumum required acceleration capacity

IO limit

Line of minimum required stiffness

0.02

0.05

0.010

0

S

p

e

c

t

r

a

l

a

c

c

e

l

e

r

a

t

i

o

n

(

g

)

0.15

0.2

0.25

0.3

0.35

0.1

0.45

0.5

0.4

Spectral displacement (m)LS limit

0.03 0.040.037

SS limit

0.05 0.06 0.080.07

Performance point ofthe initial structure

Line of target displacement

Desired performance point

at desired performance

=5%

=10%

=20%

=30%

Nonlinear dynamic

Plastic shear deformation of plate Interstory drift

PIN INERD (2.11) Linear static q factor Linear dynamic

Ultimate limit state Material strenght

Steel Elastic perfect-plastic

Ramberg - Osgood

Columns and beams

Nonlinear static behavior Verbindung Typ B

0

200

400

600

800

1000

0 10 20 30 40 50Verformung [mm] (positiv wenn Platten auf Druck)

L

a

s

t

[

k

N

]

Monotoner Druck (FEM)

Nonlinear dynamic behavior

INERD Connection "TYPE D" - Allowance for Bauschinger effect

-700

-350

0

350

700

-40 -30 -20 -10 0 10 20 30 40Displacement [mm] (positive when eye-bars in compression)

A

x

i

a

l

F

o

r

c

e

[

k

N

]

Experimental ResultsFEA Cyclic ResultsFEA Monotonic Results

Analytical formulas Design rules

y lim

Py

Pu

P

I

II

III Point I:Yield Strength ("y")

Point III:Ultimate Strength ("u")

2D or 3D conventional concentric braced frames

Pinned connection

With one or two eye bars

Nonlinear static Nonlinear dynamic

Plastic deformation (rotation) Interstorey drift of RC members (MRF)

- 16 -

BRB Buckling restrained braces (2.7) Linear static q factor Linear dynamic

Limit state criteria - Strenght - Rigidity - Stability

Steel - restrained yielding segment most of the

elastic and all of the plastic deformations take place here - restrained non-yielding segment an extension of the yielding segment but with enlarged area to ensure elastic response - unrestrained non-yielding segment used to connect the BRB to other structural elements Physical models for: Elastic global analysis Nonlinear global analysis

Nonlinear dynamic analysis

Axial-resisting and flexural-resisting mechanisms

2D or 3D conventional bracing frames Configuration: concentrical, chevron, inverted V

Pinned connection (usually); semi-rigid Connection details r.c. frame structure

Design criteria - Brace connections are to be designed with sufficient overstrength

Nonlinear statci - Pushover Nonlinear dynamic - Time-history

Core plastic deformation Interstory drift

EBF Eccentric braced frames (2.8, 3.7) Linear static q factor Linear dynamic

Limit state criteria - Strenght - Rigidity - Stability

Steel Analytical formulas Behavior (short, long link) Numerical models

Eccentric bracing frames configuration Horizontal or vertical link

Types of connection Pinned or semirigid connection

Nonlinear statci - Pushover Nonlinear dynamic - Time-history

Link plastic deformation Interstory drift

Steel jacketing reinforcement (3.4) Composite column Analytical formulas (design metodologies) for - increase ductility of column - increase column resistance to axial force - ductility index - axial force ratio

2D and 3D moment resisting frames Linear static

- 17 -

FRP (3.3) Linear static q factor Linear dynamic

Ultimate limit state Material strenght

Linear behavior (fiber + matrix)

Analytical formula Flexural strengthening Changing the failure mode from shear to flexure Avoiding lap-splice failure Confinement

2D and 3D moment resisting frames Bonding structure

Failure modes

Mode 1 - Laminate/sheet end debonding; Mode 2 - Intermediate debonding, caused by flexural cracks; Mode 3 - Debonding caused by diagonal shear cracks; Mode 4 - Debonding caused by irregularities and roughness of concrete surface; Bonding length

Nonlinear static Nonlinear dynamic

Incresed plastic deformation (rotation) Interstorey drift

DISIPATIVE DEVICE Magneto-rheological device (2.6) - Reinforced concrete frames

- Steel frames - Iron frames - Timber frames

Columns Beams Braces

Frames with dampered braces Linear static for pre-design phase Ultimate limit state Strenght and rigidity demand Ultimate displacement

S

t

r

u

c

t

u

r

a

l

t

y

p

o

l

o

g

y

- Masonry shear walls - Reinforced concrete shear walls

Shear walls

Controlable fluid damper

Parametric and nonparametric models Mechanical model Rheological models

Wall structures with shock absorber

The dievice is pin connected into the structure

Design phase Nonlinear dynamic

Plastic deformation of members Interstorey drift Specific device design parameters (range of displacement, frecvency)

18

D10 Set-up of analytical models for special materials and special devices for the seismic structural control

1. Models and performance criteria for structural elements of different material A complex problem like evaluating the bearing capacity of a structure and to decide on the necessity of a structural intervention must start from the study and thorough understanding of the real nature and stress-strain behavior of materials and elements behavior. This chapter of the report treats the most common structural typologies of monumental and historical buildings and the component materials like: masonry, marble and lime-stone, iron, reinforced concrete and timber.

1.1. Masonry elements Clay brick This section treats especially the behavior of the masonry structures made by clay brick and cement mortar. In the Mediterranean aria exists a wide range of masonry elements in terms of pattern and also in term of component materials, like adobe, stone with lime mortar etc. All this typologies of masonry respect, more or less, in terms of behavior the same principles.

1.1.1. Introduction This section was prepared in accordance with data-sheet no. 9-1 Simplified and Advanced Models for Calculation and Analysis of Masonry Shear Walls provided by A. Dogariu, T. Nagy-Gyorgy, C. Daescu, D. Daniel, V. Stoian and D. Dubina from Politehnica University of Timisoara (ROPUT). Masonry is the oldest building material (Figure 1.1-1) that still finds wide use on today building industries and remains one of the most used construction material. The most important advantage of masonry construction is its simplicity, but in the same time a lot of difficulties arise in evaluating its behavior. When a masonry building is the subject of strengthening, the strength technique cannot avoid considering the base material behavior, and modelling. The document makes a review from the analytical formula of masonry behavior, as a composite material, in different loading condition thru advanced numerical possibilities. Also it is focus on shear behavior of masonry panels only, without treating the out-of-plain behavior, and elements like arches and vaults. This problematic is treated in detail in Chapter 4 of this report.

Figure 1.1-1 Brick making in Egypt (wall painting in the tomb of Rekhmara at Thebes 1500 BC).

1.1.2. General problems of the material and element behavior The main problem of masonry is that it is a nonhomogeneous material composed of two materials with different mechanical properties. The equivalence with a homogenous material is difficult to be done because of the dependency of masonry on several factors: properties of the materials, typology of masonry, quality of workmanship etc.

One of the main problems is the great vulnerability of the masonry to earthquake due to the lack of resistance (small tensile resistance), small deformability and low ductility having a sudden and brittle

19

failure. On another hand the small ratio between resistance and own weight of the material (the masonry elements are massive with a great mass and rigidity) attract high inertia forces.

The great variability of the masonry typology can be a great inconvenience in establishing a concise methodology to describe the characteristics of masonry behavior in terms of mechanical properties. In different parts of the world and in different historical periods masonry has known a wide application from stone elements, independent or linked with earth based material to clay brick with or without mortar. In our days the brick unit that is use for masonry elements can be classified as: solid, perforated unit, hollow unit, cellular unit, horizontally perforated unit etc. The mortar can be classified as: general purpose mortar, thin layer mortar and lightweight mortar. In the following figure (Figure 1.1-2) some of the different types of masonry elements may be observed. In the frame of PROHITECH project have been studied experimentally and numerical different masonry typologies (i.e. adobe, sale-stone with lime-mortar and clay brick). This document will focus on clay brick with the remark that the general principles presented remain the same for all the other typologies also.

Figure 1.1-2 Examples of different kinds of stone masonry, (a) rubble masonry and (b) coursed

ashlar masonry, and (c,d) possible cross sections (Lourenco, 2001) From the following scheme (Figure 1.1-3) which shows the failure modes of a masonry structure when an earthquake occurred, it may be observe that due to the direction of the loads the masonry panel demonstrates a different behavior and different failure modes. A global analysis procedure (i.e. ultimate limit state), including main failure modes collapse mechanism and evaluation of bearing capacity, is presented in detailed in Chapter 4 of this report.

Figure 1.1-3 Masonry structure behavior under earthquake action

From this point the masonry panel failure modes may be divided in two categories:

20

In plane behavior

Out-of-plane behavior This document will focus in describing the behavior and the possibility to modeling the in-plane behavior. Subjected to in plane loads a masonry panel can fail in one of the following ways (Figure 1.1-4), depending on wall geometry and load conditions:

Figure 1.1-4 In-plane failure modes of shear panels (Marzhan, 1998)

Sliding failure (a)

Flexural failure (b)

Shear failure (c) Typical in-plane failure modes for a masonry wall with openings is shown in Figure 1.1-5:

Figure 1.1-5 Critical failure modes in a masonry wall with openings, IAEE/NICEE (2004).

- Sliding failure is defined as the horizontal movement of entire parts of the wall on a single brick layer, vapor barrier or mortar bed.

- Flexural failure, when the wall behaves as a vertical cantilever under lateral bending and, either cracking in the masonry tension zone (opening of bed joints) or crushing at the wall toe.

- Shear failure is characterized by a critical combination of principal tensile and compressive stresses as a result of applying combined shear and compression, and leads to typical diagonal cracks. In practice, two types of shear cracking can be observed, joint cracking by local sliding along the bed joint and diagonal cracking associated with cracks running through the bricks as well as the joints.

All this failure types should be considered when we try to determine the elements resistance in case of design or check of a structure. The design codes for masonry offer for each failure type precise formulas for resistance associated with a failure mode.

21

1.1.3. Models for masonry component materials (Material model) In case of a composite material like masonry the first step for calculation and analysis requires an exact study of the component material. Our document will describe the behavior of clay brick and cement mortar. Material test on single brick blocks and mortar specimens (Figure 1.1-6) are needed in order to establish the behavior of each component.

Figure 1.1-6 Compression tests for brick/mortar units

Usually, this test offers a behavior curve from which we can extract the load bearing capacity and deformation characteristics. These tests give us the physical models for each material. In Figure 1.1-7 the physical models for brick and mortar in case of uniaxial load are presented.

Figure 1.1-7 Behavior curves for brick and mortar

Some conclusions about material behavior can be summarized:

The component material have higher resistance in compression than in tension;

In general brick has higher elastic modulus and resistance than mortar;

In comparison with brick, which have a brittle failure, the mortar has a higher ductility and admits higher deformations.

1.1.4. Models for masonry as a composite material (Element model walls) In this paragraph the possibilities to take in account a masonry wall subjected at in plane load will be summarized. The ways to design and analyze an element are presented in a formally ranked order from simple formulas to complex numerical approaches.

a) Design code relationship (Analytical Models) From the practical point of view of design, the national standards offer different formulas for each failure mode. For example, the old Romanian Code for Masonry Building P2/85 and EC6 suggests simple mathematical relations (Table 1.1-1) for establishing the resistance of an element.

Table 1.1-1 Design formulas

22

P2/85 EC6

Sliding failure

Small eccentricity

0( 0.7 )iCF fi

AT R f

= + (1.1-1)

High eccentricity

00.7 iCF

i

f AT

= (1.1-2)

1 01( ) (1 )

cosRd vd p pF zu f l t

= + (1.1-3)

0.07 4 1pp

hl

= (1.1-4)

Flexural failure

0

0

1.25

1.25

c cCM

c c

c

M Ne RSTZ Z Z

S A eNA

R

= = =

=

=

(1.1-5) 2 342 ( ) 0.8 cos bRd d st p pz

EF zu f I h tE

= (1.1-6)

Shear failure

01 0.8p iCPi p

R AT

R

= + (1.1-7) 0

3 ( ) 0.6cosvd p p

Rd

f l tF zu

= (1.1-8)

The minimum value from associated resistances gives the most possible failure mode and the correspondent capacity.

Similar relations are given in most standards for masonry buildings available in the world.

This kind of approach is the easiest one, but gives information only about the resistance of the element, resistance that usually is highly under evaluated.

Some other codes offer more detailed information about masonry behavior, like constitutive laws, rigidity characteristic, deformation capacity, performance criteria etc.

The following figure (Figure 1.1-8) presents qualitatively the uniaxial behavior of the masonry compared with the component materials.

23

Figure 1.1-8 Behavior curves for masonry and his components (uniaxial loading) and simplified tri-linear stressstrain model for masonry Hemant (2007)

Based on component mechanical properties, Hilsdorf proposed a relation to determine the masonry properties:

( )0.9( )

4.1

234.5

b bt mw

bt b

m

b

m

f f ffU f f

hh

fU

+=

+

=

=

mf - mortar compression strength

bf - brick compression strength

btf - brick tensile strength

mh - mortar joint depth

bh - brick depth

U non-uniform stress distributing factor

b) Simplified relations for numerical analysis considering masonry as an homogenous and isotropic material (Analytical Models) The formulas available are determined from experimental tests on sub-assemblages (Figure 1.1-9):

Compression test Shear test

Figure 1.1-9 Usual tests on sub-assemblages i. Uniaxial loading

Different empirical equations are suggested in scientific papers and standards as constitutive laws for masonry elements as a homogenous material (Table 1.1-2).

Table 1.1-2 Constitutive laws

Turnsek-Cacovic (1970): 1.17

6.4 5.4k k k

=

(1.1-9)

Sawko (1982) based on Powell-Hodgkinson experimental test:

2

2k k k

=

(1.1-10)

ANDIL (Italian Association of Clay Brick Producers):

EC 6 propose : 2

2 0

1

kk k k

k uk

for

for

= =

(1.1-12)

Legend:

24

0.5

3.4142* 1 1k k

= + (1.1-11) k = maximum allowable compression strength

k = 0.002 (characteristic strain correspondent k )

u = 0.003 0.0045 (ultimate strain)

ii. Biaxial loading

Being a nonhomogeneous material with different component materials, masonry has a very different behavior in relation to direction of the applied load. Consequently the resistance of the element is highly dependent of the biaxial state of stress in the element.

Failure mode for different biaxial tests specimens are presented in next table (Table 1.1-3):

Table 1.1-3 Different biaxial tests Dhanasekar et al. (1985)

Many experimental works (Page, 1981) have been carried out and they suggest the follow biaxial interaction curve between the principal stresses (Figure 1.1-10).

Figure 1.1-10 Experimental interaction curves

25

Based on experimental tests a failure domain (Figure 1.1-11 Concrete smeared cracking used by ABAQUS) used for numerical, finite element simulation can be established. Also, other types like Hill type + Rankine type composite yield surface, Hoffman type single yield surface, yield surface proposed by Dhanasekar (1986) or by Ganz (1989)

Figure 1.1-11 Theoretical interaction curves (ABAQUS)

iii. Shear loading

The most important phenomenon that governs the behavior of masonry panels is shear behavior. In order to determine the mechanical characteristics needed in analysis some easy tests (Figure 1.1-12) should be performed on double (a) or triple (b) samples. The testing set-up is presented bellow:

Figure 1.1-12 Pure shear test

Tests at different levels of normal pre-compression showed that the failure criteria have a Mohr Coulomb shape.

26

Figure 1.1-13 Mohr Coulomb

behavior

The Mohr Coulomb relationship is described by the following formula:

0ult v = + (1.1-13) 0 - ultimate tangential stress at zero level of pre compression (cohesion)

- friction coefficient v pre compression stress

a) Test set-up b) Applied forces

Figure 1.1-14 Apparatus to obtain shear behaviour (van der Pluijm 1993) Other more sophisticated failure criteria like DruckerPragerCap modified or other models that catch the influence of normal state of stress at the ultimate shear capacity, can be used as well.

Knowing the geometry of a panel and the loads applied, the principal stress can be determined with the following relationships:

0w

VA

= the average compressive stress due to vertical load

w

HA

= the average shear stress due to lateral load H

2 2

2 2cb

= + +

0 0 (1.1-14)

2 2

2 2tb

= +

0 0 (1.1-15) Aw the horizontal cross section of the wall

b the shear stress distribution factor depending on the geometry of the wall

c) Physical models for masonry infill panels truss equivalent elements (Numerical Models) Basic Features of Masonry Walls Modeling It is one of the most used ways of modellation masonry bracing walls, and consists in replacing the masonry panel thru a linear brace element (Figure 1.1-15). Using this technique global analysis of the building with masonry infilled walls in both elastic and plastic domain can be performed.

27

,max minm

Sliding failureV Compression failure

Diagonal tension failure

=

Figure 1.1-15 Main characteristic of the system Constitutive Law Values for each of the resistance Vm (Stafford Smith 1962, Mainstone 1974, Klingner & Bertero 1978 and FEMA 306) can be established from design standards and scientific literature.

Sliding failure , 0.6s

m TS ltV = (1.1-16)

Compression failure , 0m s mV l t N = + (1.1-17) 2

0 0.04

m m

m

uN E l th

f

= =

Diagonal tension failure , cosm cr mV atf = , (1.1-18)

0.4

2 2

1/ 4

0.175( )

sin 24

m

m m m

m

c g m

a h d

d l h

E tE I h

=

= +

=

Figure 1.1-16 Equivalent diagonal strut (Al Chaar, 2002)

Panagiotakos & Fardis, 1994

28

Like all the others models this one considers the masonry panel like a strut with a specify rigidity and resistance. This model has the advantage of taking into account empirically the openings by reducing the strut width. The constitutive law of the equivalent element is defining by the follow relation:

0

20 cos

m m m

m

m m

m

G t IkhE t ak

d

=

=

(1.1-19)

,

,max ,1.3m y ms m m

m m y

V f t lV V

=

=

(1.1-20)

msf - shear strength according to diagonal compression test

Constitutive Law

Mostafaei & Kabeyasawa, 2004

,max minmSliding failure

VCompression failure

=

Constitutive Law

cosm w

mdu

=

2 2ww w w

w

harctg d l hl

= = +

,max ,max 00 , ,max ,2 ; 0.20 ( 1.33 )1

m m mm y m m y

m

V V k uk V V V

u

= = = =

(1.1-21)

29

Al-Chaar, 2002

Constitutive Law

Geometrical parameters are defined:

coscolumn

column

al

= (1.1-22) - costanm

columncolumn

ah

l = (1.1-23)

Openings and existing infill damage are considered by reducing the diagonal strut width

2

11 2

2

0.6

306

open

red panel

ARa aR R A

R Table FEMA

= = =

(1.1-24)

The presented evaluation procedures are applicable to all building structures that have been constructed with RC frames and walls that consist of infill panels constructed of solid clay brick, concrete block, and hollow clay tile masonry. In the case of old building structures the evaluation must be changed.

These assumptions are done in order to avoid dealing with the complicate behavior of masonry walls and are covering the life safety requests from the codes. The numerical analyses become easier.

This kind of supposition allow the user to perform both elastic and plastic analyzes, but without observing the state of stress distribution inside wall element or local damages, being more appropriate for global structural analysis.

d) Advanced Modeling of masonry In the last forty years an enormous growth in the development of numerical tools for structural analysis has been achieved. Historical structures are particularly difficult to be analyzed due to the lack of data. Nevertheless, significant information can be obtained from numerical analysis.

Today, the finite element method is usually adopted to achieve sophisticated simulations of the structural behavior. A mathematical description of the material behavior, which yields the relation between the stress and strain tensor in a material point of the body, is necessary for this purpose. This mathematical description is commonly named a constitutive model and an important objective of todays research is to obtain robust numerical tools, capable of predicting the behavior of the structure from the linear elastic stage, through cracking and degradation until total loss of strength.

Continuous modeling of masonry The first step toward carrying out such analyses is to develop adequate constitutive models. For masonry elements, basically three approach levels have been addressed (Rots, 1991) (Figure 1.1-17):

30

Micro-modeling where units are represented by continuum elements whereas the behavior of the mortar joints and unit-mortar interface is lumped in discontinuous or interface elements. A complete micro-model must include all the failure mechanisms of masonry, namely, cracking of joints, sliding over one head or bed joint, cracking of the units and crushing of masonry.

In the micro-model each component of masonry unit, mortar (simplified), and unit/mortar joint (detailed) must be represented with different finite elements. The employment of a micro-model to analyze an entire building becomes prohibitive, since it would result in a large number of finite elements, and consequently require a lot of computer resources to run the analyses.

Two approaches can be used: the first one is the simplified or layer model, without taking into account the interface (friction law) between brick unit elements and mortar elements (Figure 1.1-17a), and the second one detailed or interface model, by introducing a normal and tangential contact surface instead of mortar layers (Figure 1.1-17b).

These kinds of detailed and simplified micro-model have very accurate results in case of suitable input data. This type of analysis is the most advanced level of numerical simulation in case of masonry elements. It is very suitable for simulating out-of-plane behavior of masonry, but for in-plane behavior this type of approach is not justified due to the high complexity compared with similar results as in easier approaches.

However if there is a high interest in observing local behavior and interaction with other elements or material this technique may be the only one that leads to coherent results.

Macro-modeling where an anisotropic continuum model establishes the relation between average stresses and average strains in masonry.

Units and joints are not represented anymore and the geometry of masonry constituents (units and joints) is lost (Figure 1.1-17c). An adequate macro-model must include anisotropic elastic and inelastic behavior.

Figure 1.1-17 Advances modeling approach (a) masonry sample; (b) detailed micro-modeling; (c)

simplified micro-modeling; (d) macromodeling (Lourenco, 1996)

31

This type of analysis is the most suitable form the point of view of balance between involved time and accuracy of the results. Anyway macro-modeling require an extra process, homogenization introduced by Salamon (Salamon, 1968). Homogenization of masonry step that has been widely treated in articles proposing complicated energy and deformation compatibility equations. Even so, the obtained results must be seriously calibrated after this homogenization, in order to obtain a good correlation with the experimental tests.

The homogenization process, proposed by (Pande, 1998) (Figure 1.1-18), in two steps has as results an elastic orthotropic material representing the anisotropic behavior of masonry.

[ ] 1

1 0 0 0

1 0 0 0

1 0 0 0

10 0 0 0 0

10 0 0 0 0

10 0 0 0 0

xy xz

x x x

yx yz

y y y

zyzx

z z z

xy

yz

xz

E E E

E E E

E E EE

G

G

G

=

Figure 1.1-18 Homogenization steps The homogenization process described previously has many weak points as:

it is suitable only in elastic range,

it does not take into account the real pattern of masonry wall, the results being the same for the next figure (Figure 1.1-19):

Figure 1.1-19 Pattern of masonry wall (a) stack bond; (b) stretcher bond. (Lourenco, 1998)

Other methods of homogenization are presented below (Gang, 2006) (Table 1.1-4)

Table 1.1-4 Homogenization method exemple

Methods of Homogenization 1E (MPa) 2E (MPa) 12v 12G (MPa)

FEM, Stack bond (Anthoine, 1995) 8530 6790 0.196 2580

FEM, Running bond (Anthoine, 1995) 8620 6770 0.200 2620

Periodic Model, Stack bond 8568 6850 0.191 2594

32

Periodic Model, Running bond 8574 6809 0.197 2620

Multilayer Method (Pande et al. 1989) 8525 6906 0.208 2569 Two-step Method (Pietruszczak & Niu 1992) 9187 6588 0.215 2658

Elliptical Cylinder Model (Bati et al., 1999) 7784 6315 0.247 2556

The most modern way of homogenization is proposed by P. B. Loureno and A. Zucchini (Lourenco, 2002) and is based on extracting from the element a basic-cell (Figure 1.1-20). Other authors have chosen different cells.

Figure 1.1-20 Basic cell (Lourenco, 2002)

For the purpose of understanding the internal deformational behavior of masonry, detailed finite element calculations were carried out for different homogeneous loading conditions (Figure 1.1-21).

Figure 1.1-21 Basic cell components and behavior (Lourenco, 2002)

Writing the simple equilibrium equation for below scheme (Figure 1.1-22) the elastic modulus of the homogenous material in the direction of loading can be obtained.

33

Figure 1.1-22 Basic cell stress conditions

Roberto Capozucca and Fabrizio Collini developed, based on Lourenco theory a homogenization technique for analysis of a shear wall. They studied a panel of small width s compared with the dimensions of the wall in the x-y plane.

The wall is considered to be stratified with the thickness of layer hi. The continuum is considered transversally isotropic as a result of the symmetry around the vertical axis y. The stress and strain relationships for the homogeneous continuum are evaluated considering the equivalence of energy of stratified element, Ur, with the energy of homogeneous element U0 (Capozucca, 2002).

Ur = U0

The energy of the stratified element is expressed as follows:

12 i

Tr i iV

iU dV = Other more simple technique for homogenization is proposed. Starting from the experimental global behavior we can extract the rigidity of the element and obtain the elastic modulus. After establishing an initial value for elastic modulus and compressive ultimate stress of the material, the numerical simulation and calibration of the model in order to obtain a good fit of the experimental results and numerical simulation can be obtained.

31

12

Kh hEI GA

=

+

2(1 )EG

=

+

E

34

2u

cEf =

Appling a constitutive law

Discontinuous modeling of masonry (Tzamtzis, 2003) Recently a considerable attention has also been given to rational assessment methodologies, which deal more directly with the discontinuous nature of structural masonry.

The discontinuities in continuous systems are in fact interfaces between dissimilar materials and joints or fractures in the material. A survey of the literature on finite element modeling of cracks and joints shows that two main approaches are common for a representative analysis: the discrete crack and smeared crack approach and the use of joint or interface elements. Discrete crack approach Discrete crack models explicitly represent the crack as a separation of nodes (Figure 1.1-23). When the stress or strain at a node, or the average in adjacent elements, exceeds a given value, the node is redefined as two nodes and the elements on either side are allowed to separate. While this produces a realistic representation of the opening crack, a coarse meshing in the finite element model may result in misrepresentation of the propagating crack tip. A more serious problem is that, changing the formulation of the finite element model changes the number of equations to be solved and broadens the bandwidth of the stiffness matrix.

Figure 1.1-23 Discrete crack

Smeared crack approach In the smeared crack approach, cracks and joints are modeled in an average sense by appropriate modifying the material properties at the integration points of regular finite elements.

Smeared cracks are convenient when the crack orientations are not known beforehand, because the formation of a crack involves no re-meshing or new degrees of freedom. However, they have only limited ability to model sharp discontinuities and represent the topology or material behavior in the vicinity of the crack.

The smeared crack concept, based upon strain decomposition and first developed for use in concrete structures, has also been extended to the analysis of masonry elements. The method is attractive if global analysis of large-scale masonry structures is required. It does not make a distinction between individual bricks and joints, but treats masonry as an anisotropic composite such that joints and cracks are smeared out. An inherent limitation of the smeared crack approach is that discrete cracks are smeared out over an entire element and crack opening is modeled by the continuous displacement approximation functions of the conventional finite element approach (Figure 1.1-24). In view of this limitation, as well as other problems such as mesh-dependency due to tensile and compressive softening and difficulties of model calibration, smeared crack models should only be used with caution for the analysis of discontinuous structures.

35

Figure 1.1-24 Smeared crack

Interface smeared crack approach an interface smeared crack model that combines the advantages of the discrete and smeared approaches described above is proposed.

The model treat