1

Archi, pareti, volte:quali modelli per le costruzioni in muratura?

Piranesi: Pantheon (Choisy)

Luigi Gambarotta

Dipartimento di Ingegneria delleCostruzioni, dell’Ambiente e del TerritorioUniversità di Genovagambarotta@dicat.unige.it

ad Alfredo Corsanego

Historic masonry constructions: from damage to safety

1. Knowledge about historic constructions:• Historical investigation• Constructiontechniques and materials

• Survey-damage

2. Mechanical modeling:

• Simulation

• interpretation of damage –

diagnosis

• Safety evaluation

• Evaluation of strengthening

techniques

3. Design

• Retrofitting (if required)

• Monitoring

V. Lamberti, Statica degli edifici, Napoli, 1781

2

Arches

Umbria-Marche Earthquake, 1997

Masonry bridges

Road bridgeArquata S., Alessandria

PrestwoodBridge (Page, 1993)

3

Masonry walls Out-of-plane collapse

Umbria-Marche Earthquake, Colfiorito, 1997

Masonry walls In-plane collapse

Umbria-Marche Earthquake, Colfiorito, 1997

4

Vaults

South Piedmont Earthquake, 2003

Umbria-Marche Earthquake, 1997

S. Pietro Dome in RomaMichelangelo - Della Porta e Fontana, 1590

•Boscovich, Le Seur, Jacquier, 1743

•Poleni, 1748 – Vanvitelli•Burri, Beltrami, Di Stefano, Como

5

Collapse mechanism“Tre Matematici”

Boscovich, Le Seur, Jacquier,

1743

Statically admissible stress fields

Least abutment thrust,Como, 1998

Elastic NTR solutionComo, 1998

Equilibrium analysis, Poleni1748

Ancient building construction techniquesand rules of practice

Curioni, 1885

Rondelet, 1817

6

In the absence of rules……

S. Cristoforo Castle, Piedmont

Irregular distribution of masonry walls…….

7

The masonry construction

• Construction vs. Structure

• Interaction among vaults, walls, columns, arches, etc.

• Building – foundation interactions

• Modification and extension of the construction during its life

• Building to building interaction (Historic centers and urban aggregations)

Modeling: general aspects

The masonry material• heterogeneous material (periodic – random bond pattern)• components: brick unit, stone block, mortar layer• quasi-brittle behavior• different types of bond pattern – thick masonry walls• large variability of mechanical parameters• to be calibrated by in situ & laboratory tests• constitutive modeling based on the geometry and pattern of the

components and their constitutive models

Experimental response of the constituents

Masonry pillars: Stress-strain resultsand statistical summary(Schueremans & van Gemert, 2006)

Shear test apparatus - Triplet(Binda et al., 1995).

mean shear stress - mean shear strain γτ

mean normal extension - ε

σBrick / mortar

Brick-mortar interface

8

Attuatori

Cyclic shear test set up (Anthoine et al., 1994)

Slender wall

Hysteretic behavior of shear walls

Squat wall

Imposed horizontal displacement on compressed walls

b=100cmh=135cm

b=100cmh=200cm

Hysteresis & damage Dominant NL elastic response NTR

Modeling: introductory aspects

Elasticity

Unilateral contact

Plasticity

Friction

Damage

Fracture

Viscoelasticity

The constitutive ingredients

9

Geometry of the components and bond patterns

Order Disorder

Masonry heterogeneity – length scales

Structural scale Mesoscopic scale Microstructural scale

S1 40m≈ −ℓ

M

M

- brick masonry

10 25

- stone masonry

15 75

cm

cm

≈ −

≈ −

ℓ

ℓ

m1mm≈ℓ

brick

mortar

10

Homogenization

Periodic bond pattern masonry

Attuatori

Homogeneous macro-strain

Strain localization

RVE

Macro S,ES,ES,ES,E

meso s,es,es,es,e

Alpa & Monetto, JMPS, 1994, Anthoine, JSS, 1995 ………………………………

Compressive strength (solid brick masonry)

brick

mortar

masonry

Generalized hingeat a masonry arch

11

RVE FE Models

Massart et al., 2004 -Lourenco et al, 2006…

Modeling compressed solid brick masonry

c

m

t

mc

b

t

b

t

m

t

b

M

f

f

f

f

fff

+

+=

α

α

Equivalent layered mediumHilsdorf, 1969, Francis, 1971…

a=hb/hm

hb - brick unit thicknesshm - mortar layer thickness

x

zy

sz(<0)

hb

hm

+

+

+

+

+

sx

sxb(>0)

sxm(<0)

D

e = 0 e = 4cm e = 6cm e = 8cm

1 unit stack 2 unit stack

u

3. Columns and arches

Eccentrically loaded columns & arches

Experimental set up

12

0

50

100

150

200

250

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

[mm]

[kN]

e=0

e=4

e=6

e=8

Spostamento

Carico

Displacement

Load

δ ≅ 1.4-1.5

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.2 0.4 0.6 0.8 1.0

N/N0

M/M0

Brittle Duct.: 1.25 Duct.: 1.5

Duct.: 2 Ductile Exp. Genoa-1

Exp. Genoa-2 e/h=1/6 e/h=1/5

e/h=1/4 e/h=1/3 e/h=2/5

e/h=1/8 e/h=1/15 e/h=1/2

Hatz. et al. Maurenbr. Drysd. Ham, 1979

Drysd. Ham, 1982

Experimental limit domains N-M

A

A

B

B C

C

D

D

1/2

1 − N/No

M/Mo

Brencich e Gambarotta, 2005, 2008

Theoretical limit domain N-M

NTR + EPP compression

Eccentrically loaded columns & arches

Main issues

Failure of eccentrically compressed masonry pillars (solid brick masonry)

• Influence of the materials: constitutive models of brick-mortar joint -interfaces

• Free edge effects (comparison with layered models)

• Stress concentration/singularity due to mismatch of material parameters

• Influence of the brick bond / geometry of the units

........., b m m

l t c t

b b b b

c c c c

b a t, ,

d b

N f f fF , , , ,

f d f f bfb

e =

• Validity limits of 2D models

N

N

e

ab

t

d

RVE

13

N

M

l

N

M

N

M

l

N

M

l

0

h /2a

yb

y

h /2b

0x

xB

a

Bb

S

h

2a

h

2b

x

y

l

ha

h

2b

h

2b

Bb

Ba

Bb

Periodic cell

REV

symmetry

( ) ( ) ( ) ( ) ( )0 0 0

1 1

, aMN

a p p d a

nm n m

n m

x y a f x r f x a f x g y= =

Φ = + + ∑∑

( ) ( ) ( ) ( ) ( )0 0 0

1 1

, bMN

b p p d b

nm n m

n m

x y a f x r f x b f x g y= =

Φ = + + ∑∑

Assumed tension field

+ BC’s on f() and g()

+ plastic admissibility - Mohr-Coulomb criterion

+ unilateral – frictional brick-layer interface

max

T

eq

att

N = ≤ = ≤

c a

S a d

A a 0

A a 0

ɶPPLIN

Eccentrically loaded columns & arches

LB strength of stacked bond prisms

Influence of the RVE height/width ratio

x

y

x

y

M=0

N

130h mm=

55 d mm=

x

y

x

y

M=0

N

65h mm=

120 d mm=

x

y

x

y

M=0

N

65h mm= 250 d mm=

xxσ

yyσ

xyσ

0.4

0.6

0.8

1.0

0 50 100 150 200 250

h [mm]

N0 / f

cbh

extended layered medium

present model

0.88 -

d (mm)

Concentric axial force

14

Eccentric axial force

Vertical compressive stressat the limit state

0e =

12

he =

6

he =

4

he =

3

he =

[N/mm2]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.2 0.4 0.6 0.8 1.0N / N 0

M / M

0

present model

homogeneous beam

e = h

/3

e = h/4

e = h/6

e = h /12

N-M limit domain

NTR-EPP homogeneous beam

• No influence of the friction coefficientin the range [ ]0 0 6, .µ∈

( )2

0 0 0

1, 0

2

M N Nf N M

M N N

= + + =

Limit domain of NTR-EPP (NoTensileResistant – ElasticPerfectlyPlastic)

homogeneous beam

2

0 0 4 4c

MM N d f d= =

0 c

MN f d=

3. Masonry arch bridges

ε

σ

σ0

ε el

Incremental analysis – Castigliano2D - Homogeneous beam model

G

σ'c

σc

no-tensile resitant area

MN

compression

h

x

yuni-axial model

NRT+ EPP compression

Brencich et al, 2003

15

Limit analysis - NTR modelHypotheses:

1. No tensile resistance NTR 2. Unbounded compressive strength

3. No sliding failure admitted 4. Small displacement and rotations

Trust line

µs f

b0

c s max .=µ µµ µµ µµ µSafe theorem

Statically admissible stress fields

Kooharian, Heyman, ………..

G

G

G2 2

'

'

1

C1

'

C1

G1

−θ11θ

−θ2θ2

2C 2C'

s

kµ q

u+

u−

1θ sµc

uv

( ) ( )k hb u s ds q u s ds v v= −∫ ∫ɺ ɺµ γµ γµ γµ γS S

Kinematic theorem c k min=µ µµ µµ µµ µ

Kinematically admissible mechanisms

rh 165=

sh 220=

1428f =

6550=ℓ

a 1 4= ℓb 300=

2000′=ℓ 2000′ =ℓ

Masonry bridges:Vault – fill interaction

Prestwood Bridge hinges

Live load

Tests on full scale masonry bridges: Prestwood Bridge

Page, 1993

Heavy not resisting fill

P 46kNu =

P 228kNexp =

16

(Royles & Hendry, 1991)

Tests on model scale bridges

Vault and fill

Vault

Complete bridge

v ( mm )

Total load applied

F ( kN)

cFfill

Fc

c

c

F4

F

fill

≅≅≅≅

Crisfield (1985)

Choo et al. (1991)

Bicanic et al. (2003)

Owen et al. (1998)

Sperimentazione su modelli – Effetto del riempimento - A. Brencich

17

Sperimentazione su modelli – Effetto del riempimento - A. Brencich

Two dimensional model of the bridgeTwo dimensional model of the bridge

Spandrel walls

Fill

Fill

Vaults

PierBacking

Fill: frictional-cohesive material

Arches and piers:NTR, ductile in compression beams

The two-dimensional model isobtained by neglecting the in plane resistance of the spandrels

Why a two-dimensional model ? - Focus on longitudinal collapse mechanisms and arch-fill interaction

- Difficulties in describing the 3D behaviour of materials and components

Limits of validity - Interaction between longitudinal and transverse collapsemechanisms(within the objectives of

the research)

18

1x2x

3x

Spandrelwalls

Fill

13 23 0τ τ= =

Assumptionsabout the fill:

c 33 0σ−σ ≤ ≤ɶ

,3 0ijσ =

Effects of spandrel walls on fill resistanceEffects of spandrel walls on fill resistance

The effects of spandrels and tie-rods on the fill resistance(due to the containing effect) are approximately taken intoaccount by limiting the out-of plane stress

33σ

Spandrel walls

2x

3x

33σ 33σ

Arch barrel

The plane state conditions necessary to obtain a two-dimensional description of the fill implyan underestimation of its resistance (safe assumptions for the fill resistance)

The resistance of the fill is affected by the containingcapacity of the spandrels, which has to be taken intoaccount also in a plane model of the fill

Method of analysis: Limit Analysis approach Method of analysis: Limit Analysis approach

Limit Analysisapproach

Lower Bound

Upper Bound

Equilibrium model

(C & G, 2007)

Compatible model

(C & G, 2005, 2006)

Assumptions (and limits)- Associated flow rule

- Ductile behaviour

- Small displacements

Why Limit Analysis ? - Reduction of the number of constitutive parameters withrespect to an incremental analysis (reduction of uncertainties)

- Direct evaluation and description of the collapse configuration

- Evaluation of a range of values within which the true collapsload of the mechanical model is contained

A range of values isobtained within whichthe true collapse load of the mechanicalmodel is contained

19

Admissible domains: arches and beamsAdmissible domains: arches and beams

( )b b

p p p

M 2N Nf 1 0

M N N

= + + ≤

σσσσ

p cN bh= σ2

p c

1M bh

4= σ

pM M

pN N

1 2−1−

cσ

N

M

NTR, ductile in compression beam

Admissible domain

Associated flow rule

( )Sign M;

hχ= λɺɺ

p

1 2N1

2 N

ε= + λ ;

ɺɺ λ≥0ɺ

1 2

( ) ( ) ( )21 2

mc 11 22 12 11 22f 4 sin 2ccos 0σ σ τ σ σ= − + + + ϕ− ϕ ≤σσσσ

( ) ( ) ( ) ( )22 2

mc 11 22 12 11 22 3

2 sin4ccosf 4 sin 0

sin sinσ σ τ σ σ σ

1− ϕϕ= − + + + ϕ − − ≤

1+ ϕ 1+ ϕσσσσ

( ) ( ) ( ) ( )23 2

mc 11 22 12 11 22 3

2 sin4ccosf 4 sin 0

sin sinσ σ τ σ σ σ

1+ ϕϕ= − + − + ϕ− − ≤

1− ϕ 1− ϕσσσσ

( ) ( ) ( )2 2

t 11 22 12 11 22f 4 0σ σ τ σ σ= − + + + ≤σσσσ

max

3 3 cσ = σ ≥ −σɶ

max

3 1 2

1 sin 2ccosmin( , )

1 sin 1 sinσ σ σ − ϕ ϕ

= + =+ ϕ + ϕ ( ) ( )2 2

11 22 11 22 12

1 sin 2ccos1 1 42 2 1 sin 1 sin

σ σ σ σ τ − ϕ ϕ + − − + + + ϕ + ϕ

( ) ( ) ( )2 2

mc 11 22 11 22 12 c

1 sin 4ccosf 4 2 0

1 sin 1 sinσ σ σ σ τ σ + ϕ ϕ

= − + + − + − − ≤− ϕ − ϕ

ɶ ɶσσσσ

Admissible domains: fill (1)Admissible domains: fill (1)

(1)

(2)

(3)

(Mohr-Coulomb criterion in the plane of the model)

(Containing stress delimitation)

(Tension cut-off)

Maximum allowed containing stress from spandrels

- angle of internal friction

c

ϕ- cohesion

cσɶ - Allowed containing stress (from spandrel)

Constitutive parameters:Mohr-Coulomb criterion, out-of-plane components:

Maximum admissible

out-of-plane stress 3σ

(Minimum thrust on

the spandrels)

3 cσ ≥ −σɶ

20

1σ

2σ

2ccos 1+ sin

1- sin 1- sinc

ϕ ϕ− − σ

ϕ ϕɶ

2ccos 1+ sin

1- sin 1- sinc

ϕ ϕ− − σ

ϕ ϕɶ

2ccos

1- sin

ϕ−

ϕ

2ccos

1- sin

ϕ−

ϕ

1

mcf 0=

tf 0=

mcf 0=ɶ

Resulting admissible domain of the fill in the

space of in-plane principal stresses:

Plane stress

assumption

( )0c

σ =ɶ

Admissible domains: fill (2)Admissible domains: fill (2)

(Boundary due to the limited containing capacity

of spandrels; it does not exist in plane strain )

(Tension cut-off)

(In-plane Mohr-Coulomb)

Admissible domains: archAdmissible domains: arch--fill interfacefill interface

Slidings between arch and fill are

allowed and ruled by Coulomb law

with tension cut-off:

( )( )

c

n

t

n

f tan c 0

f 0

σ τ

σ

= ϕ+ − ≤

= ≤

σσσσ

σσσσ

nσc

-cϕ

τ

Tension cut-off

(Tension cut-off)

- angle of friction

c

ϕ- cohesion (assumed zero)

Constitutive parameters:

21

FE FE discretizationdiscretization of the bridgeof the bridge

Interface

elements

Triangular

elements

Beam elements

Discontinuity allowed between

triangular elements

EquilibriumEquilibrium model: model: beambeam elementelement

ξξξξ

νννν

i j

Fiν

iFξC i

F jν

Fj

ξ

Cj

ξ

( )t ξ

( )m ξ

b

F

F

C

r

r r

r

ξ

ν

=

s( )

( )( )b

N

M

ξξ

ξ

=

σσσσ

( ) ( ) ( )2

T

b

1 1N F 1 0

2 2

=− + − + −

bi

ξ

ξ ξ ξξ τ τ ξ ξξξξl l l

l l l

( ) ( ) ( )

( ) ( )

2

2 2

2 3

2 2 2 T

b

1 h 1 1 1M C F 1 0 1 0

2 2 2 3

1 h 1 1,

4 6 2

=− − − − + − +

− + −

b

i i

ν ν

ξ ξ ξ ξξ ξ τ σ

ξ ξτ σ ξ νννν

l ll l l l l

l l l ll l l

( ) ( ) ( ) ( )b b b b b b

i i h h k kξ ξ ξ ξ+ + += D s D D dσ σ σσ σ σσ σ σσ σ σ

h

k

-t

i

jh

k

t

νννν

ξξξξ

νννν

ξξξξ

νννν

ξξξξ

1 2

1 2−1−

p

M

M

p

N

N

bf 0=k+

bf 0=k-

bf 0=

( ) ( )T

b b b bξ ξ= − ≤f N r 0σσσσ

s

1

p 1t

tξ

−=

−l ( )T

b b b b ,t

tξ= − ≤f N r 0σσσσ s1...pt=

Internal force vectorfrom equilibrium

The discrete admissible domain is inside the original one in order to maintain the lower bound property of the load multiplier

22

CompatibleCompatible model: model: beambeam elementelement

h

k

-t

i

jh

k

t

νννν

ξξξξ

νννν

ξξξξ

νννν

ξξξξ

1 2

1 2−1−

p

MM

p

NN

ɺεεεε

α

β

1∆ ɺuα

2∆ ɺuα

2∆ ɺϑ1∆ ɺϑ

ɺϑi

ɺϑ j

ɺ iuβ

ɺ juβ

ɺ juα

2

1

i j

ɺ iuα

ɺmuα

m

e e e= B uɺ ɺεεεε e

i

i

i

j

j

j

m

u

u

u

u

u

α

β

α

β

α

ϑ

ϑ

=

u

ɺ

ɺɺ

ɺ ɺ

ɺɺ

ɺ

e eT e e= − ≤f N r 0σσσσ e e e= N ɺɺε λε λε λε λ

1

1e

2

2

u

u

α

α

ϑ

ϑ

∆ ∆= ∆ ∆

ɺɺ

ɺɺɺ

εεεε

e e e

b− =B u N 0ɺɺ λλλλ

e T e

p bD = r ɺλλλλ( )( )

be

b

0

l

=

σσσσσσσσ

σσσσ

Compatibility

Dissipated

power

Two hinges at ends

Associated

flow rule

The discrete admissible domain is outside the original one in order to maintain the upper bound property of the load multiplier

EquilibriumEquilibrium model: model: triangulartriangular elementelement

i

j

h { }T

11 22 12

h h h hσ σ τ=σσσσ

{ }T

11 22 12

j j j jσ σ τ=σσσσ

{ }T

11 22 12

i i i iσ σ τ=σσσσ

( ) ( ) ( ) ( )11

22

12

N N Ni i j j h h

σστ

= = + +

x x x xσ σ σ σσ σ σ σσ σ σ σσ σ σ σ

( )N p q

pqδ=x

( ) 1 2Ni i i ia x b x c= + +x

11,1 12,2 1 0bσ τ+ + =

12,1 22,2 2 0bτ σ+ + =

t t t t

i i j j h h+ + + =D D D d 0σ σ σσ σ σσ σ σσ σ σ1x

2x

Discretized limitdomain

11σ

22σ

12τ

( )( ) ( )T ≤f x = N x r 0σ σ −σ σ −σ σ −σ σ −

( ) T ,k k ≤f = N r 0σ σ −σ σ −σ σ −σ σ − , ,k i j h=

( )xσσσσ linearly

interpolated

Equilibrium

Plastic admissibility

23

Upper and Upper and lowerlower boundbound collapsecollapse loadload

Equilibrium

Plastic admissibility

Dead load

Live load multiplier

0 s= +q q q

( )lb s

s 0

T

s max s ,

,

s ,

.

=

= ≤

C c

Q q q

N r

σ =σ =σ =σ =

σ −σ −σ −σ −

σσσσ

( )T T T

lb 0 3

T T

3

T

3

s min ,

,

1,

.

= + + + + =

= ≥

c q y

C Q Y 0

q

0

1 21 21 21 2

1 21 21 21 2

2222

α α αα α αα α αα α α

α α αα α αα α αα α α

αααα

αααα

Maximum statically admissibleload multiplier

Dual

Linear Programmingproblems solved by internalpoint method (Mehrotra)

Live load

( )0T T

ub

T

s min

1

= +

−

= =

q v r

Bv Ν 0

Av 0

q v

ɺ

ɺ

λλλλ

λ =λ =λ =λ =

Minimum kinematicallyadmissible load multiplier

Compatibility and associated flow rule

Boundaryconditions

Boundary conditions and internal constraints

Positivity of the power of live load

PRESTWOODPRESTWOOD BRIDGEBRIDGE

Hinge at impost

Hinge at haunch

Non Non resistantresistant fillfill model: model: collapsecollapse mechanismmechanism

P 46kNu =

4.5MPaσ =co37ϕ =

030η=

expP 228kN=

(Page, 1987)

Numerical collapseload (fill as dead load): 20% of the experimentalresult

24

maxγ 1

max 10γ −⋅ 2

max 10γ −⋅ 3

max 10γ −⋅ 4

max 10γ −⋅

(b)

(c) maxγ 1

max 10γ −⋅ 2

max 10γ −⋅ 3

max 10γ −⋅ 4

max 10γ −⋅

(b)

(c)

4.5MPaσ =co37ϕ =

expP 228kN=

PRESTWOODPRESTWOOD BRIDGE:BRIDGE: PlanePlane strainstrain fillfill

ubP 254kN=

lbP 209kN=

Compatible model:

Equilibrium model:

Line of thrust

Minimum principal stress directions

Maximum shear strainrate field

PRESTWOODPRESTWOOD BRIDGE:BRIDGE:SensitivitySensitivity of the of the collapsecollapse load load PPuu toto the the constitutiveconstitutive parametersparameters

5 10 15 20c (kPa)

0

50

100

150

200

250

300

pu (kN

)

UB

LB

Non resistant fill

Exp

5 10 15 20c (kPa)

0

50

100

150

200

250

300

pu (kN

)

UB

LB

Non resistant fill

Exp

0 2 4 6 8 10σc (MPa)

0

50

100

150

200

250

300

pu (kN

)

( )

UB

LB

Exp

.

10 20 30 40ϕ

0

50

100

150

200

250

300

pu (kN

)

LB

UB

Non resistant fill

Exp

10 20 30 40ϕ

0

50

100

150

200

250

300

pu (kN

)

LB

UB

Non resistant fill

Exp

Masonry compressive strength

Angle of internal frictionCohesion

PlanePlane strainstrain fillfill

25

0 20 40 60σ3 (kPa)

0

25

50

75

100

125

150

175

200

225

pu (kN

)

63− 30− 10−20− 0 ( kPa)

In-plane Mohr-Coulomb limit domain activation:

Containing stress distribution:

PRESTWOODPRESTWOOD BRIDGEBRIDGEEffectsEffects of the of the containingcontaining stress of the stress of the spandrelsspandrels

0 20 40 60σ3 (kPa)

0

25

50

75

100

125

150

175

200

225

pu (kN

)

20− 10−15− 5− 0 ( kPa)

In-plane Mohr-Coulomb limit domain activation:

Activation of the containing stress limit condition:

Containing stress distribution:Localisation of collapse under the live load position

PRESTWOODPRESTWOOD BRIDGEBRIDGEEffectsEffects of the of the containingcontaining stressstress

26

0 20 40 60σ3 (kPa)

0

25

50

75

100

125

150

175

200

225

pu (kN

)

Activation of the containing stress limit condition:

Localisation of collapse under the live load position

PRESTWOODPRESTWOOD BRIDGEBRIDGEEffectsEffects of the of the containingcontaining stressstress

Results obtained by avoiding plastic strains in the fillbelow the position of the applied live load: in this case the reduction of the collapse load is very small

This result shows that a diffused deep reduction of the fill resistance due to a limited containing stress from spandrels does not affect the result, once localised effects under the load are avoided

Prestwood Bridge

Load\deflection curve and ductility demand

Masonry ductility:

c

εδ =ε

Vertical displacement v

0 1 2 3 4 5 6 7 8

v/ℓ (*10-4)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Pu /(cℓ

2)

1δ =2δ =

3δ =

c = 0.01 MPaφ = 37

o

σc = 4.5 MPa

Upper boundUpper boundIncremental analysis

4(*10 )

27

Multi span bridge: Fill – arches – piers interaction

Discrete domain planes

Arch model properties:

p 36=Fill density

318 kN /mρ = Masonry density

Fill model properties:3

18 kN /mρ =Discrete domain planes p 48=

b 6=

rh 1=

sh 1=

7=f

h 10=

sh 1= =14ℓ p

h 3= =14ℓ

12MPaσ =co30ϕ =

20kPac =

Non resistant fill

P 625 kNu =

UP 2468 kN=

Multi span bridge

U.B. – collapse mechanism

L.B. – statically admissible stress field

LP 2150 kN=

28

Probabilistic models of masonry arches (including non-linear material response)

� Monte Carlo simulation + Limit Analysis (Ng and Fairfield, 2002)� Fuzzy non linear analysis (Biondini et al., 2002)� Probabilistic Limit Analysis -Hystorical Buildings (Augusti et al., 2001, 2002)

(Schueremans & van Gemert, 2006)

Masonry pillars:Stress-strain resultsand statistical summary

Uniaxial compression

M

N

Pcfɶ

( ) 2

1, , 1 0

2c

c c c

MN NN M f

bhf bhf bh f

= + + =

ɶɶ ɶ ɶ

φ

Probabilistic limit domain of the GH

N

M

fc 0,5fc 1,5fc 2fc

Effect of the compressive strength

P P2

1

F c

Nf

Mbh

N

= ≤ − +

ɶ

Probability of failure of the masonry section

Hypotheses:

� the compressive strength is a random variable

� which is correlated in the section

� the tensile strength is deterministically vanishing

depending on the random variable

Probabilistic limit domain of the generalized hinge

cfɶ

N

M

N*

CDF PDF

cfɶ

29

N

M

2q

q

j

1

Piecewise linearization of the limit domain - 2q limit lines

s

( )( )

( )2

2

1P P P

1

T

F c i ii i

if r r

q

− = ≤ = ≤ −

n sɶ ɶ

= T

i ir n s

( )1 1 0 T+ = ≤s n sφ

Probability of failure of i-th sectional mechanism nq

for i =1 the limit domain is deterministic

( )( )

2

2

1=

1i c

ir f

q

−

−ɶɶ

Projection of the internal

force on i-th mechanism

Strength of i-th mechanism

random variable

int,

T

i i i iD r= =s ε λ

N

M

( )irɶσvariance

2

int,

1

1i i i i c

iD r f

q

−= = −

ɶɶ ɶλ λ

2

int,

1

1i i i i c

iD r f

q

−= = −

λ λ

( ) ( ) ( )2

int,

1

1i i i i c

iD r f

q

−= = −

ɶɶ ɶσ λσ λ σ

Arch discretization + Selection of the set of all possible mechanisms

i

j

hk

sf

( )( )44 4 !2

4! 4 !m n q

nN C R q

n= =

−

Kinematical model of the archn potential generalized plastic hinges

Mechanism enumeration

� External Power

(deterministic) 0, , 0j j j a jW W s W= + ≥

0,

,

j j

j

a j

D Ws

W

−=ɶ

ɶ� Kinematically admissible multiplier

(random variable)

� Rate of dissipation(random variable)

4 4

1 1

T

j j p p p c p p

p p

D r f= =

= = =∑ ∑r λ ɶɶ ɶ ɶ λ ψ λ

NKANo. admissible mechanisms

Hypotheses on the compressive strengths at the n hinges:

� stochastically independent

� Gaussian-distributed (stable)with the same expected strength and variance

C j C jP E P s s = < ɶ

j-th mechanism

30

Masonry bridges: probabilistic analysis

c.o.v.

15%

6,45m

11,25m

17,70m

sP sP sP sP

1,00sP

0,665sP

1,137sP

0,935sP 0,263sP

1

2

3

4 5 6

7

8

9

450 kN 450 kN

sP

N

M

1

4,9

6

1

4

6

9

sP sP sP

1200 1250 1300 1350 1400 1450 15000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−3

s

P(s

)

U.B.

L.B.

CDF

10-3

First mechanism

Prarolo Bridge, Genoa, 40m span

[ ] j

j

E E = ∪� Collapse of the arch(realization of at least one event)

� CORNELL bounds

( ) [ ] ( )C C Cmax P P E 1j j

j

E P E≤ ≤ ≤∑good approximations for small probabilities <<10-3

c.o.v.

15%

Masonry railway bridges

Open problems

� 2D vs. 3D modeling

� Interactions among structural

elements (spandrel walls……….)

� Uncertainty of material properties

� Non linear analysis including

damage and cracking

� Dynamic response to time-

varying loadings

� Foundation settlements

� Material degradation

?