Seminari DICAT
14 Marzo, 2007
Mechanics of masonry structures:
arches, shear walls and vaults
Luigi Gambarotta
[email protected]
Layout:
•
•
•
•
•
•
Historic and old masonry buildings
Modelling: general aspects
Columns, arches and bridges
Walls
Domes
Conclusions
Web site:
prinpontimuratura.diseg.unige.it
Piranesi: Pantheon (Choisy)
1. Historic masonry constructions: from damage to safety
1. Knowledge about historic
constructions:
•Historical research
•Historic construction
techniques and materials
•Inspection-damage
2. Mechanical modeling:
•Interpretation of damages –
diagnosis
•Simulation
•Assessment
•Evaluation of strengthening
techniques
V. Lamberti, Statica degli edifici, Napoli, 1781
3. Design
•Assessment of structural
safety
•Design of repairs (if required)
Arches
Temple of Sethi I and
Ramses II, XIX Dinasty
Roma,
Mercati traianei
Palace at Ctesiphon, A.D. 550
Arches
Umbria-Marche Earthquake, 1997
Masonry bridges
Prestwood
Bridge (Page, 1993)
Road bridge
Arquata S., Alessandria
Railway bridge (Bologna-Piacenza)
Masonry walls
Umbria-Marche Earthquake, Colfiorito, 1997
Out-of-plane collapse
Masonry walls
Umbria-Marche Earthquake, Colfiorito, 1997
In-plane collapse
South Piemonte
Earthquake, 2003
Vaults
Umbria-Marche Earthquake, 1997
Pantheon
Masonry domes
S. Maria del Fiore
S. Pietro
Masonry domes
Basilica di S. Maria di Carignano - Genova
Materials and bond patterns
dis
ord
e
r
Old building construction techniques
and rules of practice
Rondelet
Curioni
Old building construction techniques
and rules of practice
In the absence of rules……
Caste in S. Cristoforo, Genova
2. Modeling: general aspects
The aims of mechanical modeling masonry constructions
• interpretation of the damage and (realistic) assessment of the
structural safety;
• selection of the most efficient and less invasive repairs and
strengthening techniques (if necessary), compatible with the
original design concepts of the construction.
Understanding the relevant mechanical behavior of the construction
through proper structural models (avoiding dogmatic conventional
assessment procedure)
Masonry
• heterogeneous material (periodic – random bond pattern)
• components: brick unit, stone block, mortar layer
• quasi-brittle behavior
• different types of bond pattern – thick masonry walls
• Randomness of the material parameters
• to be calibrated by in situ set up
• constitutive modeling based on the geometry and assembly of the
components and their constitutive models
2. Modeling: general aspects
(cont’d)
The masonry construction
• Construction Versus structure
• Mechanical interaction among the construction elements (vaults, walls,
columns, arches, ……)
• Building – foundation interactions
• Modification and extension of the construction (superfetations, growth, etc…)
• Building to building interaction (Historic centres and urban aggregates)
Other aspects
• Sensitivity to the applied loads: static (weigth loads) V/s dynamic (seismic,
traffic…) loads.
• Sensitivity to the construction sequence
• Influence of initial stresses and strains and quasi-brittle behavior of masonry:
how to approach the safety assessment?
• Chemo-physical degradation and residual life
2. Modeling: general aspects
Cyclic shear test set up
(Anthoine et al., 1994)
Imposed horizontal
displacement on
compressed walls
Attuatori
Hysteresis & damage
Dominant NL elastic response NTR
Squat wall
Slender
wall
b=100cm
h=135cm
b=100cm
h=200cm
2. Modeling: introductory aspects
The constitutive ingredients
Elasticity
Unilateral contact
Plasticity
Friction
Damage
Fracture
Viscoelasticity
Homogenization
Periodic bond pattern masonry
localization
Attuatori
Macro S,E
micro s,e
Homogeneous macro-strain
RVE
3. Columns and arches
Compressive strength
hb brick unit thickness
a=hb/hm
hm mortar layer thickness
fM =
α f bt + f mt
α
f bt
f bc
+
f mt
f mc
Hilsdorf, 1969
3. Columns and arches
Eccentrically loaded columns & arches
Experimental set up
u
1 unit stack
e=0
e = 4cm
2 unit stack
e = 6cm
e = 8cm
Corradi et al, 2006
Eccentrically loaded columns & arches
M/Mo
Theoretical limit domain N-M
250
1/2
D
NTR + EPP compression
[kN]
C
B
200
A
e=0
−1
e=4
Carico
Load
150
e=6
C
B
e=8
D
100
δ ≅ 1.4-1.5
Experimental limit domains N-M
0.30
M/M0
50
[mm]
0
0.25
0.0
0.5
1.0
1.5
2.0
2.5
Spostamento
Displacement
3.0
3.5
N/No
A
4.0
0.20
0.15
0.10
0.05
N/N0
0.00
0.0
Brencich e Gambarotta, RILEM, 2005
0.2
Brittle
Duct.: 2
Exp. Genoa-2
e/h=1/4
e/h=1/8
Hatz. et al.
Drysd. Ham, 1982
0.4
0.6
Duct.: 1.25
Ductile
e/h=1/6
e/h=1/3
e/h=1/15
Maurenbr.
0.8
1.0
Duct.: 1.5
Exp. Genoa-1
e/h=1/5
e/h=2/5
e/h=1/2
Drysd. Ham, 1979
Eccentrically loaded columns & arches
Periodic cell
M
N
Assumed tension field
b
hb
2
B
y
Φ
a
ha
B
x
hb
2
a
( x, y ) = a
p
0
⎡⎣ f 0 ( x ) + r f
p
N
d
0
( x )⎤⎦ + ∑∑ anm f n ( x ) g ma ( y )
b
B
l
Φ
simmetry
⎡⎣ f
N
p
0
Mb
( x ) + r f ( x )⎤⎦ + ∑∑ bnm f n ( x ) g mb ( y )
d
0
n =1 m =1
+ plastic admissibility
0
B
yb
b
hb /2
y
S
B
ha
2
( x, y ) = a
p
0
n =1 m =1
+ B.C. on f() e g()
x
hb
2
b
Ma
x
a
ha /2
+ unilateral – frictional brick-layer interface
0
l
⎧min N = cT a
⎪
⎪ ⎧S a ≤ d
⎪
PPLIN ⎨ ⎪
⎪
⎪ t.c. ⎨ A eq a = 0
⎪ ⎪
⎪ ⎪⎩ A att a ≤ 0
⎩
REV
N
M
l
Influence
of the unit shape ratio h/b
0.5
Concentric load
Boundary effects
b
0.4
B
y
x
M / M0
l
0.3
l=120
l=250
a
B
σ xx
E.P.
0.2
σ yy
0.1
0.0
0.0
0.2
0.4
0.6
N / N0
0.8
1.0
σ xy
Incremental analysis – Castigliano
Iterative updating of the compressed section
3. Masonry bridges
σ'c
σ'c
σc
σ
compression
y
M
x
h
G
εel
N
ε
compression
M
x
h
G
σ0
no-tensile resitant area
no-tensile resitant area
NRT + EPP compression
NRT
N
σ'c
∆NEP
com pression
P
x
150
80°
75
375
h
G
N
no-tensile resitant area
1500
1500
R=937.5
M
x
220
750
36.87°
Brencich et al, 2003
Limit analysis - NTR model
Kooharian, Heyman, ………..
Hypotheses:
1. No tensile strength masonry NTR
2. Infinite compressive strength
3. No sliding failure
4. Small displacement and rotations
Thrust line
Statically admissible stress field
µs f
Safe theorem
b0
µ c = max µ s .
Trust line
Limit analysis – NRT model
µ k = − ∫ γ hb u v ( s ) ds
Kinematically admissible mechanisms
∫ q u ( s ) ds
v
S
S
µkq
µk q
θ2
'
G2
b = γ hb
−θ 2
C'2
C2
G2
R
'
G1
G1
θ1
h
−θ 1
'
C1
'
G2
G
C1
s
'
2
+
u
uv
−
u
θ1 s
'
G1
G1
C1
'
C1
Potential failure mechanism.
µc
Kinematic theorem
µ c = min µ k
µq
Limit analysis: applications
F
C22 '
C2
Steel rod
A,σ0
Tie rod
G2
G2
G1
G1
θB
Effects of the limited
compressive strength (2° hypot.)
'
'
f (N, M )= |M| - Nh = 0
2
θ
C1
vv
Hθ
C12
C12
w
horizontal displacement rate
M
elastic - NTR
limit domain
n
Columns
n
A
C
B
Vertical displacement rate
θB
Arch
N0
El - NTR - Plastic domain
f (N, M )= |M| - Nh [1- N ] = 0
N0
2
N
Masonry bridges:
Vault – fill interaction
Tests on full scale masonry
bridges: Prestwood Bridge
Page, 1993
Pexp = 228kN
Live load
Heavy not resisting fill
b = 300
a =1 4
h r = 165
h s = 220
f =1428
′ = 2000
Pu = 46kN
= 6550
hinges
′ = 2000
Prestwood Bridge
Tests on model scale bridges
Total load applied F ( kN )
(Royles & Hendry, 1991)
Fcfill
Fcfill
≅4
Fc
Complete bridge
Vault and fill
Fc
Vault
Crisfield (1985)
v ( mm )
Choo et al. (1991)
Owen et al. (1998)
Bicanic et al. (2003)
Limit analysis of the bridge
p = pp 0 + sp
Ii
∂Ωf
Cavicchi & Gambarotta, 2005, 06, 07
Li
Ωf
u
ϑ
u
bb
Ω
b
i
b
ϑ
u
(a)
Hypotheses:
x2
bb
O
x1
(b)
1. Arches & piers : NTR – EPP in
compression
Limit domains
2. Fill: Mohr-Coulomb + Cut-off
fb = 0
f mc = 0
M Mp
12
τ 12
σ 22
N Np
1
ft = 0
12
σ 11
Fill: Mohr Coulamb + Cut off
Arch: NTR - EEP in compression
3. FE discretization
4. Plane strain/plane stress
5. Piecewise linearization of the
limit domains
Triangular
elements
ξ
u2k
ξ
t
k
u
-t
u1h
h
h
ν
ν
Equilibrium FE model
arch – fill interaction
f bk+ = 0
Lower bound
M
Mp
ft = 0
τ 12
(a)
∆u
g k+
b =0
−1
N
Np
σ 22
gt = 0
2
(σ 11 −σ 22 )
2
−1 2
g bk-
(b)
u1i
τ 12
12
N
u1j
j
Compatible FE model
Arch – fill interaction
Upper bound
∆ uα
ϑϕj j
Beam ν
elements
ν
M
i
2
ξ
u2j
i
Interface
elements
i
σc
u
ϑϕii
j
u1k
k
ξ
k
h
∆ϑ
h
2
σ 11
=0
(c)
(a)
f bk- = 0
Piecewise linearization of the limit domains
(b)
Prestwood Bridge collapse: numerical simulation
U. B. - collapse mechanism (plane strain)
γ max
γ max ⋅10−1
γ max ⋅10−2
γ max ⋅10−3
γ max ⋅10−4
Hinge at haunch
σc = 4.5MPa
ϕ = 37o
c = 10kPa
(Page, 1993)
Pexp = 228kN
Plane strain
U. B. -Collapse mechanism
PU = 228kN
(a)
γ max
γ max ⋅10−1
γ max ⋅10−2
γ max ⋅10−3
γ max ⋅10−4
L.B. – Principal stress field
PL = 184kN
(b)
Lateral pressure
−49.7
−14.9 −9.9 −5
0 ( kPa)
Plane stess
PU = 184kN
(a)
γ max
γ max ⋅10−1
γ max ⋅10−2
γ max ⋅10−3
γ max ⋅10−4
PL = 160kN
Inluence of the cohesion and the angle of internal friction on the collapse load
450
400
350
350
300
300
250
LB
200
150
200
100
50
50
10
(
20
c (kPa)
σc = 4.5MPa, ϕ = 37 o
LB NRT
150
LB NRT
0
UB NRT
250
100
0
UB
400
UB NRT
pu (kN)
pu (kN)
450
UB
LB
0
30
0
)
350
10
20
30
40
φ
( σc = 4.5MPa, c = 10kPa )
UB
UB NRT
300
Influence of the masonry
compressive strength on the
collapse load
pu (kN)
250
200
LB
150
LB NRT
100
50
0
0
5
(
10
15
σc (MPa)
c = 10kPa, ϕ = 37 o
)
20
50
Prestwood Bridge
Load\deflection curve and ductility demand
Incremental
analysis
UpperUpper
bound
bound
0.8
0.7
P u /(c 2 )
0.6
0.5
δ =1
0.4
δ=3
δ=2
0.3
c = 0.01 MPa
φ = 37o
σc = 4.5 MPa
0.2
0.1
0
0
1
2
3
4 5 6
(*104-4))
v/A (*10
7
Vertical displacement v
8
Masonry ductility:
ε
δ=
εc
Multi span bridge: Fill – arches – piers interaction
Fill model properties:
Fill density
Discrete domain planes
ρ = 18 kN / m3
p = 36
Arch model properties:
Masonry density
Discrete domain planes
ρ = 18 kN / m3
p = 48
h r =1
h s =1
σc = 12 MPa
ϕ = 30o
f =7
c = 20kPa
h =10
b=6
h s =1
=14
hp =3
=14
Multi span bridge
U.B. – collapse mechanism
Non resistant fill
PU = 2468 kN
L.B. – statically admissible
stress field
Pu = 625 kN
( Pu = 923 kN )
PL = 2150 kN
Sterpi et al., 2006
Masonry bridges: probabilistic analysis
Compressive strength: a random variable
P.D.F.
M
fc
y
N
N = f c by
M =
f c by
(h − y )
2
N
M
2m piecewise linearization
Kinematical model of the arch
n potential generalized plastic
k hinges
sf
Pr ( s ) = Prob ⎡⎣∃u ∈V : f0T u + sf T u > rT λ p ⎤⎦
h
Discrete model – failure - i-th mechanism
j
i
nt = Cn4 Rm4 =
n!
m4
4!( n − 4 ) !
Mechanism enumeration
Approximations: bounds on the C.D.F.
[ Ei ] = ⎡⎣f
T
0
ui + sf u i > r λ pi ⎤⎦
T
T
si =
Pr [ Ei ] = Pr [ si < s0 ]
rT λ pi − f0T ui
f T ui
⎛
⎞
max Pr ( Ei ) ≤ Pr ⎜ ∪ Ei ⎟ ≤ ∑ Pr ( Ei )
⎝ i
⎠ i
Masonry bridges: probabilistic analysis
450 kN
450 kN
17,70m
Prarolo Bridge, Genova, 40m span
11,25m
6,45m
sP
sP
sP
sP
5
4
6
3
7
0,935sP 0,263sP
2
8
1,137sP
1
9
0,665sP
1,00sP
sP
c.o.v.
15%
M
sP
sP sP
4,9
6
4
1
9
N
First mechanism CDF
2
6
1
−3
x 10
1.8
1.6
Hypotheses:
The compressive masonry
strength is gaussian
s=
Dint − W0
Wa
1.2
P(s)
Statistically independent random
variables
1.4
U.B.
10-3
1
0.8
λ C (r ) λ
L.B.
0.6
T
c.o.v. =
The structural strength (upper
bound theorem) is gaussian
Dint − W0
0.4
0.2
0
1200
1250
1300
1350
s
1400
1450
1500
Masonry railway bridges
Open problems
?
Non linear analysis
including damage
and cracking
4. Masonry walls – Simulation of in-plane response to seismic actions
Cyclic horizontal forces, anisotropic damage, damage localization, hysteretic dissipation,
inertial vertical forces …………
Travi di servizio
640
Cella di carico
(50 t)
Cella di carico
2+2 barre ø35 L=6150 mm, filettate
(25 t)
Martinetti
(50 t)
0.00
PROSPETTO
PROSPETTO DELLA PARETE
SEZIONE LONGITUDINALE
Block masonry wall in S.Sisto (Beolchini et al., 1997).
Brick masonry wall tested in Pavia, Magenes et al. (1994).
10
u2
u1
F2
[daN*10-3]
5
0
u [mm]
-5
-10
-10
a)
-10
-5
0
5
10
330
4. Shear wall – in-plane response
Shear testing on brick-mortar assemblages
Shear test apparatus - Triplet
(Binda et al., 1995).
τ mean shear stress - γ mean shear strain
σn
ε - mean normal extension
Experimental results
Phenomenological description
4. Shear wall – in-plane response
Direct cyclic shear test by Atkinson et al., 1989.
Brick-mortar interface model: coupled damage-frictional interface
Gambarotta e Lagomarsino, 1997
b /2
s
Macro fieds
b /2
σ m = {σ t σ n τ}t
ε *m = {0 ε *m γ *m }t
Inelastic strain
ε m = {0 ε m γ m }
t
Vm
Total strain
ε m = K mσ m + ε *m
ε *m = h(α m ) H ( σ n ) σ n
γ *m = k (α m ) ( τ − f )
Conjugate variables
am damage variable
f friction
Damage
evolution
Sliding
Ym = 12 h'( α m ) H ( σ n ) σ n2 + 12 k '( α m ) ( τ − f )2 , γ*m
φdm = Ym − Rm ≤ 0
φdm = 0, φdm ≤ 0, αm ≥ 0, φdmαm = 0
φ s = f + µσ n ≤ 0
.
.
.
γ *m = v λ , λ ≥ 0
v=
f
f
Brick-mortar interface model: coupled damage-frictional interface
b/2
s
b/2
Vm
Limit states
Slid
ing
&
dam
age
Elastic
Hysteretic damage
Simulation of experimental results (Binda et al)
4. Masonry walls – simulation of the in-plane response
FE model
Brick units + Interface
(c)
SR
OR
(b)
(a)
OR
A
BR
(a)
B
OR
C
(b)
(d)
4. Masonry walls – Discrete models
Casciaro et al, 2002
Salerno, Uva, 2006
Coupled damage-frictional
interface
(Gambarotta e Lagomarsino, 1997)
Blocchi rigidi
Mixed FE formulation
Arch-length iterative analysis
4. Large masonry shear walls – seismic actions
Micro fields σ, u, ε, ζ
Macro fields Σ, Ε, Ζ
u ( x ) = Ex + u per
divσ = 0 in E
σn antiperiodic on ∂E
σ n=0
∂E
su I
Micro – costitutive equations
I
E
σm ↔ εm , ζ m
Interface
σi ↔ εi , ζ i
ζ internal variables
1
x ⊗ tds
A ∂∫E
E=
1
sym ( u ⊗ n ) ds
A ∂∫E
Macro – costitutive equations
Σ ↔ Ε, Ζ
Brick units σ b ↔ ε b , ζ b
Mortar
Σ=
Periodic RVE
Ζ internal variables
Layered micro-model
4. Continuum damage-friction model
ε = KM σ + η ε + η ε
*
m m
*
b b
(Gambarotta e Lagomarsino, 1997)
ε *b = {0 ε b γ b }t
b /2
Brick unit
s
b /2
ε = {ε1 ε 2 γ }t
Vm
σ = {σ1 σ 2 τ}t
ε*m = {0 ε m γ m }t
Mean stress
interface
Interface
ε m = cmn α m H (σ 2 ) σ 2
Limit conditions:
γ m = cmt α m (τ − f )
• Damage
Internal variables: am damage & f interface friction
Conjugate variables
Ym = 1 cmn H ( σ2 ) σ22 + 1 cmt ( τ − f )2 , γ m
2
2
φ dm = Ym − Rm (α m ) ≤ 0
φ db = Yb − Rb (α b ) ≤ 0
• Friction
Brick unit
φ s = f + µσ 2 ≤ 0
ε b = cbn α b H (-σ 2 ) σ 2
.
γ b = cbt α b τ
Internal variavle: ab danno nel mattone
Conjugate variable
RVE
Yb = 1 cbn H (-σ 2 ) σ 22 + 1 cbt τ 2
2
2
sliding
.
.
γm = vλ , λ ≥ 0
v=f /|f |
Layered micro-model
4. Continuum damage-friction model
σ2 ≥ 0
Opened
interface
(Gambarotta e Lagomarsino, 1997)
Evolution of the internal variables
.
⎧⎪φ ⎫⎪ ⎡R' 0 ⎤⎧⎪α. ⎫⎪ ⎧⎪c σ σ. + c τ τ. ⎫⎪
m
m
dm
mn 2 2
+
.
. mt ⎬ ≤ 0
⎨ . ⎬ = −⎢
⎨
⎨
⎬
⎥
'
⎪⎭
cbt τ τ
⎪⎩φdb ⎪⎭ ⎣ 0 Rb ⎦⎪⎩αb ⎪⎭ ⎪⎩
φ dm = 21 cmn σ 22 + 21 cmt τ 2 − Rm (α m ) ≤ 0
φ db = 21 cbt τ 2 − Rb (α b ) ≤ 0
.
.
{φ
dm
γ 2m
=
− Rm ( α m ) ≤ 0
cmt α 2m
σ2 < 0
φ dm
Closed
interface
φs = τ −
1
2
γm
+ µσ 2 ≤ 0
cmt α m
φ db = 21 cbn σ 22 + 21 cbt τ 2 − Rb (α b ) ≤ 0
⎡ γ 2m
−
− Rm'
.
⎢
3
⎧φ ⎫
cmt α m
⎪⎪ .dm ⎪⎪ ⎢
vγ m
⎢
⎨ φs ⎬ = ⎢
2
⎪ . ⎪ ⎢ cmt α m
⎪⎩ φdb ⎪⎭ ⎢
0
⎢⎣
.
{φ
.
dm
.
.
}{α
φs φdb
.
}{α
φdb
m
.
0
. .
}
t
.
{α
=0
.
m
}
αb
t
≥0
⎤
0⎥ .
0
⎫
⎥⎧⎪α.m ⎫⎪ ⎧
.
.
⎪ ⎪ ⎪
⎪
0 ⎥⎨ λ ⎬ + ⎨ v τ+ µ σ 2 ⎬ ≤ 0
⎥ .
.
.
⎥⎪⎪α b ⎪⎪ ⎪⎩cbnσ2 σ2 + cbt τ τ⎪⎭
⎥⎩ ⎭
Rb' ⎥
⎦
vγ m
cmt α2m
−1
cmt α m
λ αb
}
αb
m
t
=0
.
{α
. .
m
}
λ αb
t
≥0
Limit states
Damage in the interface
and brick units
Elastic interface
Brick damage
4. Large shear walls – simulation of experimental results
Crack pattern
(Magenes et al)
exp
DRIFT 0.1%
simul
Cyclic response of the door wall: a)
experimental; b) numerical simulation.
DRIFT 0.3%
4. Large shear walls – dynamic response to ground motion
(a) Acceleration response spectrum of the input base
motion. Amplification function with respect to the base of
the wall: (b) first floor displacement, (c) second floor
displacement.
(a) Acceleration time history applied at the base of the wall.
(b) Displacement time history on the second floor.
Cyclic response of the large scale wall: (a)
second floor; (b) first floor.
4. Large shear walls – response to horizontal forces Brencich etal, 2001
266
1586
Masonry building in Catania
GNDT
1320
Horizontal forces
superimposed on
vertical dead loads
s=57 cm
PARETE 3
PARETE 3
340
185
980
PARETE 1
s=86 cm
PARETE 2
Via G. Oberdan
Via G. Oberdan
125
360 485
310
795
1586
Via L. Capuana
Via L. Capuana
360
Cortile interno
Anteriore al 1840
Anteriore al 1840
PIANO TERRA
211
211
130
341
168
509
205
714
130
189
903 1033
193
1226
1226
.02 .05
.07 .10 .13
.15
.18
.20
.23 .25 .28
.30
.33
.35 .38
.40 .43 .45
.48
.50
.55 .55 .58
.60
.66
(a)
Deformazione angolare globale (%)
110
.53
(b)
90
80
70
60
Taglio alla base (t)
100
(c)
PUNTO 1
PUNTO 2
(d)
50
40
PUNTO 3
30
(e)
20
Taglio alla base / Peso verticale complessivo
120
.68
.68
.63
.57
.51
.46
.40
.34
.28
.23
.17
.11
.06
10
Spostamento in sommita' (cm)
0
0.4 0.8 1.2 1.6
2.0
2.4 2.8
3.2
3.6
4.0
4.4
4.8 5.2
5.6
6.0 6.4
6.8
7.2
7.6 8.0
8.4
8.8 9.2
9.6 10.0 10.4 10.8
Simplified collapse mechanism
(Como e Grimaldi)
4. Large shear walls – simplified approaches
λ F3
P2
P1
P4
P3
F3
λ F2
G2
G1
G3
G4
l3
P2
P1
λ F1
P3
P4
hp
T1
λ F3
M3i
T2
T3
T3i
i
i
T3D
T3S
F3i
λ F2
i
F2
h3
G1
G2
G3
i -1
T2D
i
T2Si
G4
T2D
Gi
h3
λ F1
h2
i
F1
i
T 1S
i
T1D
h2
h1
h1
bi
i +1
T2S
T4
4. Shear walls – influence of the unit shape and bond pattern
Experimental results
Dry block masonry
(Giuffrè et al.)
Collapse mechanisms
and limit slope angle b
λ=
For varying:
•Bond pattern
•Wall slenderness
b
l=tanb
4. Shear walls – continuum models
homogenization of elastic brick and damaging interfaces
Luciano e Sacco, 1997
4. Shear walls – Multiscale limit analysis – influence of the bond pattern
Admissible macro-stress fields (Suquet, 1983)
q0
sP
sP
Macro S,E
micro s,e
q0
S hom
• Alpa Monetto, 1994, Alpa,
Gambarotta et al 1996
De Buhan, De Felice, 1997
S b, S m unbounded, S i Coulomb
Lower bound
⎧ sL = max ( ss ) ,
⎪
⎪CΣV = c,
⎨
⎪QΣV − ss q = q 0 ,
⎪ YT Σ ≤ y.
V
⎩
• Milani et al, 2005
S b, S m Mohr-Coulomb – cut-off
S i not active
sP
⎧
⎪
⎪
⎪
⎪⎪
= ⎨Σ
⎪
⎪
⎪
⎪
⎩⎪
1
⎧
⎫⎫
x
t
Σ
=
ds
⊗
⎪
⎪⎪
A ∂∫E
⎪
⎪⎪
⎪divσ = 0 ∀x ∈ E
⎪⎪
⎪⎪
⎪⎪
⎪⎪
=
∀
∈
n
0
x
σ
I
⎨
⎬⎬
⎪σn anti-periodico su ∂E ⎪⎪
⎪
⎪⎪
α
α
S
∈
∀
∈
α
x
x
σ
E
,
=b,
m
⎪ ( )
⎪⎪
⎪
⎪⎪
i
⎪⎩σ ( x ) ∈ S ∀x ∈ I
⎭⎪⎭⎪
Dual kinematic definition of Shom
q0
FE discretization
• Equilibrium model
Upper bound
(
)
⎧ sU = min ( sk ) = min −q T0 I a + z T λ ,
⎪
⎪Ba - Z T λ = 0
⎪
⎨ Aa = 0,
⎪ T
⎪q I a = 1,
⎪
⎩λ ≥ 0 .
sP
q0
• Compatible model
Anderheggen e Knöpfel, Sloan &
coworkers, Pastor et al., Maier &
coworkers…. ……
4. Shear walls – Multiscale limit analysis – influence of the bond pattern
Homogenized failure surface
Milani et al., 2005
Collapse mechanism (U.B.) Catania Building
Brencich et al, 2000
4. Shear walls
•
In-plane model
non-local continuum model able to take into account the scale effect unit
size/structure/size, high gradients of the micro-stress field, regolarization
of damage model
Besdo, Műhlhaus, Rizzi, Trovalusci, Masiani, Salerno…..
Trovalusci e Masiani, IJSS, 2005
• Out-of-plane models
Elastic models
Cecchi e Sab, 2002, 2004,
Limit analysis:
Discrete models:
Orduna e Lourenco, 2005
Continuum models:
Sab, 2003, Milani e Tralli, 2005
4. Shear walls
• Out-of-plane collapse - multiscale models
Cecchi et al., 2006
Dissipation Power
Internal forces
Shearing Mechanisms
Flexural & Torsional Mechanisms
Elementary deformations of the Representative
Volume Element
4. Shear walls
• Out-of-plane collapse - multiscale models
Cecchi et al., 2006
Out-of-plane Collapse
Perforated shear wall
Collapse Mechanism
5. Domes
S. Pietro Dome in Roma
Poleni
Michelangelo
Della Porta e Fontana, 1590
Boscovich, Le Seur, Jacquier, 1743
Poleni, 1748 – Vanvitelli
Burri, Beltrami, Di Stefano, Como
Statically admissible
stress fields
Collapse mechanism by the “Tre
Mattematici”
Least abutment thrust,
Como
Elastic NTR solution
Como
Dome-drum interaction: Basilica di Carignano in Genova (G. Alessi, 1540-1600)
Crack pattern in the inner dome (from below)
Basilica di Carignano: Safe theorem
Statically admiddible states
Hypotheses
• NTR material
• Infinite compressive strength
• No sliding failures admitted
Equilibrium of a slice
Loads:
• masonry weight γ=17 kN/m3
• lantern weight P=1200 kN/16
Search for thrust surfaces
lying within the masonry
Lantern weight distribution
for the safe equilibrium state:
85% inner shell
15% outer shell
Gambarotta et al., 2002
P
Upper Bound Theorem
If ∃ u ∈ KinAdm such that:
W=
∫
B
-
b •u − dv +
∫
b •u + dv =Wa + Wres ≥ 0
The structure will collapse
B+
(Romano e Romano, Romano e Sacco, Como)
b - unit volume weigth
u + - upward velocity
u − - downward velocity
1. Local mechanism
Inner and outer domes
η1 =
Wres
Wa
≈ 2 >1
W <0
Overall Mechanism domes-drum int.
Global
mechanism 1.
H
η2 =
Wres
Wa
≈ 8.5
B
F
A
Global
mechanism 2.
D
η3 =
Wres
Wa
≈ 1.8 ÷ 7
E
C
Influence of the column compressive strength
on the location of the centre of rotation of the
drum slice
FE Model –1/8 slice
Problemi costitutivi
Smeared crack
localizzazione
Concentrazioni di tensione
Hypotheses:
•elastic isotropic model
•smeared cracking
•crushing
•small displacements
[10-1 MPa]
SHELLS:
• elements
• nodes
DRUM:
• elements
• prismatic
• tetrahedral 3907
• nodes
TOTAL:
• elements
• nodes
• d.o.f.s
4056
5922
17474
13567
7904
21530
13826
35000
Incremental analysis 1.5
Crushing sc=6MPa
av/g
1.25
1
C
B
A
?
0.75
0.5
0.25
Vertical displacement of the lantern [cm]
0
0
Crack pattern
State A
0.2 0.4
0.6 0.8
(inside)
1
1.2
1.4 1.6
1.8
(outside)
6. Influence of the construction sequence – structural growth
6. Influence of the construction sequence – structural growth
Construction sequence
considered
σx
σx
Brown & Goodman,
Gravitational stresses in accreted
bodies, 1963
σy
sx
sy
Gravity loads applied to
the final configuration
σy
6. Influence of the construction sequence – structural growth
d
λf
du
λ
Ωλ
βλ
u
ndλ
u(s,β;λ)
u(s,β+dβ=λ+dλ;λ+dλ)
λ+dλ
λ
λ+dλ
ndλ
u(s,β;λ+dλ)
Ωλ
λ
u(s,β;λ)
0
P
λ
β
P
Ω0
n
α
P0
λ
Q
ndλ
β
α
ey
x
ex
R0
P
λ=β
ey
(a)
x0
y
ex
Structural displacement rates
ey
ex
x
dragged displacement rates
(a)
Reference domain
β
E ( s, β ; λ ) = sym ( ∇u0 ( s, β ) ) + ∫ sym ( ∇g ( s, β ; λ ) ) γdλ +
0
Strain field
(
)
+ sym ( ∇u ( s, β ) ) + γsym ( g ( s, β ; λ = β ) − g ( s, β ; λ = β ) ) ⊗ ∇β +
λ
+ ∫ sym ( ∇g ( s, β ; λ ) ) γdλ.
β
Stess field
Bacigalupo et al., 2007
T ( s, β ; λ = λ f ) =
λf
( E ( s, β ; λ = λ ) − E ( s, β ; λ = β ) ) = ∫ sym ( ∇ ( g ( s, β , λ ) ) ) γdλ
β
f
t
(b)
6. Influence of the construction sequence – structural growth
Normal stresses at springing
Example: Triumphal arch
0.80
M.C.C
λf
M.S.C1
M.S.C2
0.60
λ
h
λ
0.40
n
2b
β
0.20
Growth
ignored
ey
s
0.00
R0
π/3
ββ ==00
1.00
-0.40
0.50
Growth
included
0.00
ϑϑ == 30
-0.20
-0.50
θ
ex
-1.00
α
t
-1.50
P
0
σ ϑϑ ( MPa )
Displacement field
Tangential component
β ( m)
ϑ ( deg )
Bacigalupo et al., 2007
7. Problems & prospects
• Discrete & Continuum models:
regular versus random masonry pattern (thickness??, real masonry);
homogenization: size effect –> unit – RVE – wall size;
interface model: brick unit – mortar layer interaction;
cohesion: strain localization, non-unique incremental solution
•Damage-frictional models seem to be necessary to understand the masonry
wall response to orizontal varying forces. What is the role of
perturbations to the reference state due to settlement, construction
sequence etc?
• NTR based model are simple and efficient when static loads inducing
moderate axial forces are considered. Can comparable simple models
be found for high compressive axial forces and time varying
loads?
•The fill and spandrel walls notably increase the load carrying capacity of
arches and masonry bridges: how this effect can be simply included in
assessment procedures?
• Incremental analysis (the reference state often is not well described) or
Limit analysis (masonry is far from to be ductile)?
• What simplified procedures for the seismic assessment of buildings and
bridges?
• Mechanical decay in the long term.
• etc. etc……………………………………