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MONTE CARLO ANALYSIS FOR THE BLAST RESISTANCE DESIGN AND ASSESSMENT OF A REINFORCED CONCRETE WALL
Advances in computational methods for resilient structural systemsunder extreme hazards
Minisymposium Organizers: J. Ricles, T. KaravasilisChair: J. Ricles
Pierluigi OlmatiP.E., Ph.D. Student
Sapienza University of RomeEmail: [email protected]
Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
2 Presentation outline
1 Introduction
2 Component damage levels and response parameters
3 Blast scenario and target
4 Fragility curves
5 Conclusions
Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
3
General view of Ronan Point prior to demolition/photo 1987/photographer
M Glendinning
Features:- apartment building,- built between 1966 and 1968,- 64 m tall with 22 story,- walls, floors, and staircases were made of precast
concrete,- each floor was supported directly by the walls in
the lower stories, (bearing walls system).
References: NISTIR 7396: Best practices for reducing the potential for progressive collapse in buildings. Washington DC: National Institute of Standards and Technology (NIST), 2007.
Ronan Point – May 16, 1968
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Cause Damage Pr. Collapse
Features:- apartment building, built between ‘66 and ‘68,- 64 m tall with 22 story,- walls, floors, and staircases were made of precast
concrete,- each floor was supported directly by the walls in
the lower stories, (bearing walls system).The event:- May 16, 1968 a gas explosion blew out an outer
panel of the 18th floor, - the loss of the bearing wall causes the progressive
collapse of the upper floors,- the impact of the upper floors’ debris caused the
progressive collapse of the lower floors.
Ronan Point – May 16, 1968
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LOAD STRUCTURE RESPONSETruck bomb1.8 ton TNT
A. P. M. BuildingBefore 19/05/95
A. P. M. BuildingAfter 19/05/95
HAZARD COLLAPSE RESISTENCE
P[●]: probabilityP[●|■]: conditional probabilityH: HazardLD: Local DamageC: CollapseNISTIR 7396
UFC 4-023-03
References:
EXPOSURE
VULNERABILITY
ROBUSTESS
∑i = P[C] P[LD|Hi]P[Hi] P[C|LD]LOCAL EFFECTCAUSE GLOBAL EFFECT
Collapse probability
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Pierluigi OlmatiSapienza University of Rome
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6 Presentation outline
1 Introduction
2 Component damage levels and response parameters
3 Blast scenario and target
4 Fragility curves
5 Conclusions
Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
r
Φelastic
Φplastic
Mplasticδ
δel
-r
-rel
Rel = rel A
R = r A
L
L δtmδe
Tension membrane effect (tm)
PlasticElastic
δlim
7
θ=arctg( 2δmax
L )μ=δ max
δ e
Response parameters
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Component damage levels θ [degree] μ [-] Blowout >10° none
Hazardous Failure ≤10° none Heavy Damage ≤5° none
Moderate Damage ≤2° none Superficial Damage none 1
Blowout: component is overwhelmed by the blast load causing debris with
significant velocities.Hazardous Failure: component has failed, and debris velocities range from
insignificant to very significant.Heavy Damage: component has not failed, but it has significant permanent
deflections causing it to be un-repairable.Moderate Damage: component has some permanent deflection. It is generally
repairable, if necessary, although replacement may be more economical and aesthetic.
Superficial Damage: component has no visible permanent damage.
Component Damage Levels
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Pierluigi OlmatiSapienza University of Rome
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9 Presentation outline
1 Introduction
2 Component damage levels and response parameters
3 Blast scenario and target
4 Fragility curves
5 Conclusions
Pierluigi OlmatiSapienza University of Rome
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Stre
et
Level 2
Level 3
Level 1
Target
10 Blast scenario - Areal view
Explosive
weight
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11 Blast scenario - Section view
Fence barrier
Vehicle bomb
w [kgp]
p [W]
Stand-off distance
r [m]
p [R]
Cladding wall
θi
p [Θi]1
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Fence barrier
Vehicle bomb
w [kgp]
p [W]
Stand-off distance
r [m]
p [R]
Cladding wall
θi
p [Θi]
Blast scenario - Section view
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13 Precast cladding wall panel
Panel dimensions:3500x1500x150 mm(137x59x6 in.)
Panel reinforcement:12 φ10 mm (0.4 in.)
Panel materials:Concrete fcm=35 MPa (5000 psi)Steel B450C (≈GR60)
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14 Input data
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Symbol Description Mean COV Distribution fc Concrete strength 28MPa 0.18 Lognormal fy Steel strength 495 MPa 0.12 Lognormal L Panel length 3500 mm 0.001 Lognormal H Panel height 150 mm 0.001 Lognormal b Panel width 1500 mm 0.001 Lognormal c Panel cover 75 mm 0.01 Lognormal
W Explosive weight 227 kgf 0.3 Lognormal R Stand-off distance 15 m 20 m 25 m 0.05 Lognormal
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Pierluigi OlmatiSapienza University of Rome
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15 Presentation outline
1 Introduction
2 Component damage levels and response parameters
3 Blast scenario and target
4 Fragility curves
5 Conclusions
Pierluigi OlmatiSapienza University of Rome
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16 Fragility curves – Failure probability
P f (X>
x0|
IM)
Intensity Measure (IM)
P f ( X>x0 )=∫− ∞
+∞
P f ( X >x0∨ℑ ) p ( ℑ ) d ℑ
p(IM
)
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CDL (j)
Z=i
MC analysis
FC-CDL (i, j, k)
FC-CDL (j,k)
FC-CDL (k)
i=N ?
j=M ?i=
i+1
j=j+
1 YES
NO
NO
YES
• CDL: Component Damage Level• R: Stand-off distance• Z: Scaled distance• FC-CDL: numerical Fragility Curves
of the Component Damage Level• i: the i-th point, of the j-th FC-CDL
corresponding to the k-th R• j: the j-th CDL• k: the k-th stand-off distance• MC analysis: Monte Carlo analysis• N: number of FC-CDL points, or
number of the Z• M: number of the CDL• L: number of the stand-off
distance• Interpolated FC-CDL: lognormal
interpolated Fragility Curves of the Component Damage Level
R=k
k=L ?
YES
NO
k=k+
1
FC-CDL
Lognormal Interpolation
Interpolated FC-CDL
j=1 i=1 k=1
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INTENSITY MEASURE
Fragility curves – Flowchart
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• CDL: Component Damage Level• R: Stand-off distance• Z: Scaled distance• FC-CDL: numerical Fragility Curve
of the Component Damage Level• i: the i-th point, of the j-th FC-
CDL corresponding to the k-th R• j: the j-th CDL• k: the k-th stand-off distance• MC analysis: Monte Carlo
analysis• N: number of FC-CDL points, or
number of the Zs• M: number of the CDLs• L: number of the stand-off
distances• Interpolated FC-CDL: lognormal
interpolated Fragility Curve of the Component Damage Level
Pierluigi OlmatiSapienza University of Rome
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ta to t-o
Pso
P-so
Po
Reflected pressure
Incident pressure
Prα
P-rα
P (t )=P r(1−tt d
)e− βt
td t a ≤ t ≤ t d
Intensity measure
Peak pressure
Impulse density
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0
20
40
60
80
100
0 0.004 0.008 0.012 0.016P
ress
ure
[kP
a]Time [sec]
R=15 m - W=20 kgp
R=30 m - W=20 kgp
R=10 m - W=20 kgp
R=20 m - W=50 kgp
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P s 0=1.772( 1
Z3 )− 0.114( 1
Z2 )+0.108( 1Z )
i0=300( 1Z
3√𝑊 )
Z=R
3√WScaled distance
Side-on pressure
Side-on impulse density
Pr=2 P s 0( 7 Patm+4 P s 0
7 Patm+ Ps 0)
t d=2i s 0
P s 0
P (t )=P r(1−tt d
)e− βt
td t a ≤ t ≤ t d
Shock duration
Shock wave
Reflected pressure
INTENSITY MEASURE
Intensity measure
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1
10
100
1000
100 1000 10000 100000
P [
kPa]
i [kPa ms]
θ=2 �θ=5 �θ=10 �
I
D
P
I: impulsive regionD: dynamic regionP: pressure region
Intensity measure
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0
20
40
60
80
100
2.4 2.6 2.8 3.0 3.2 3.4
P f(X
> x 0
|Z)
Z
Hazardous Failure1
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Intensity measure
Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
CDL (j)
Z=i
MC analysis
FC-CDL (i, j, k)
FC-CDL (j,k)
FC-CDL (k)
i=N ?
j=M ?
i=i+
1
j=j+
1 YES
NO
NO
YES
• CDL: Component Damage Level• R: Stand-off distance• Z: Scaled distance• FC-CDL: numerical Fragility Curves
of the Component Damage Level• i: the i-th point, of the j-th FC-CDL
corresponding to the k-th R• j: the j-th CDL• k: the k-th stand-off distance• MC analysis: Monte Carlo analysis• N: number of FC-CDL points, or
number of the Z• M: number of the CDL• L: number of the stand-off
distance• Interpolated FC-CDL: lognormal
interpolated Fragility Curves of the Component Damage Level
R=k
k=L ?
YES
NOk=
k+1
FC-CDL
Lognormal Interpolation
Interpolated FC-CDL
j=1 i=1 k=1
22 Fragility curves – Flowchart
Fragility curves for n° M CDLs and the k-th
stand-off distance (R)
Fragility curves for n° M CDLs and n° L stand-off
distances (R)
Fragility curve for the j-th CDL and the k-th stand-off
distance (R)
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• CDL: Component Damage Level• R: Stand-off distance• Z: Scaled distance• FC-CDL: numerical Fragility Curve of the
Component Damage Level• i: the i-th point, of the j-th FC-CDL
corresponding to the k-th R• j: the j-th CDL• k: the k-th stand-off distance• MC analysis: Monte Carlo analysis• N: number of FC-CDL points, or number
of the Zs• M: number of the CDLs• L: number of the stand-off distances• Interpolated FC-CDL: lognormal
interpolated Fragility Curve of the Component Damage Level
Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
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Fence barrier
Vehicle bomb
w [kgp]
p [W]
Stand-off distance
r [m]
p [R]
Cladding wall
θi
p [Θi]
(1) R=R0 W=W1 Z=Z1
(2) R=R0 W=W2 Z=Z2
(3) R=R0 W=W3 Z=Z3
……..(N) R=R0 W=WN Z=ZN
Z
1 2
3
NP(X>
x|Z)
Fragility curve for the j-th CDL and the k-th stand-off distance (R)
Monte Carlo Simulation
Fragility curves – Computing the fragility curve
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0
20
40
60
80
100
2.4 2.6 2.8 3.0 3.2 3.4
P f(X
> x 0
|Z)
Z
Hazardous Failure j-th CDL
k-th R
i-th Z
Fragility curves – Results
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Component damage levels θ [degree] μ [-] Blowout >10° none
Hazardous Failure ≤10° none Heavy Damage ≤5° none
Moderate Damage ≤2° none Superficial Damage none 1
0
20
40
60
80
100
2.4 2.6 2.8 3.0 3.2 3.4
P f(X
> x 0
|Z)
Z
Hazardous Failure
0
20
40
60
80
100
2.8 3.0 3.2 3.4 3.6 3.8 4.0
Heavy Damage
P f(X
> x 0
|Z)
Z
0
20
40
60
80
100
3.0 3.5 4.0 4.5 5.0
P f(X
> x 0
|Z)
Z
Moderate Damage
0
20
40
60
80
100
5 6 7 8 9 10 11
P f(X
> x 0
|Z)
Z
Superficial Damage
CDL
R
Fragility curves – Results
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Fence barrier
Vehicle bomb
w [kgp]
p [W]
Stand-off distance
r [m]
p [R]
Cladding wall
θi
p [Θi]
Blast scenario - Section view
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0
20
40
60
80
100
3.0 3.5 4.0 4.5 5.0
P f(X
> x 0
|Z)
Z
Moderate Damage
Fragility curves – Results
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SafeUnsafe
Example
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Fence barrier
Vehicle bomb
w [kgp]
p [W]
Stand-off distance
r [m]
p [R]
Cladding wall
θi
p [Θi]
𝐙=𝐑
𝟑√𝐖
Scaled distance
p [Z
]
Z
Blast scenario - Section view
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0
20
40
60
80
100
2.4 2.6 2.8 3.0 3.2 3.4
P f(X
> x 0
|Z)
Z
Hazardous Failure
p(Z)
[-]
P ( X>x0 )=∫− ∞
+∞
P f ( X>x0∨Z ) p ( Z ) dz≅∑i=0
∞
P f ( X>x0∨Z )i p ( Z )i ∆ Z i
R=Zm3√W m=Rm
Fragility curves – Failure probability
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CDL Mean W=227 kgf COV=0.3 lognormal distribution
R, COV=0.05 lognormal distribution
FC analysis MC analysis Difference Δ% R = 20 m
SD 100.0 % 100.0 % 0.0 % MD 96.6 % 97.5 % 0.9 % HD 55.7 % 55.5 % 0.3 % HF 13.6 % 12.1 % 11.0 %
R = 25 m SD 100.0 % 100.0 % 0.0 % MD 74.6 % 77.3 % 3.5 % HD 14.2 % 12.6 % 11.2 % HF 1.02 % 1.02 % 0.0 %
R = 15 m SD 100.0 % 100.0 % 0.0 % MD 97.9 % 99.9 % 2.0 % HD 93.6 % 96.9 % 3.4 % HF 67.8 % 72.6 % 6.6 %
Fragility curves – Failure probability
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Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
31 Presentation outline
1 Introduction
2 Component damage levels and response parameters
3 Blast scenario and target
4 Fragility curves
5 Conclusions
Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
32
1- Fragility curves can be helpful in the design of precast concrete wall panels, or cladding panels in general.
Conclusions
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50
20
40
60
80
100
3.0 3.5 4.0 4.5 5.0
P f(X
> x 0
|Z)
Z
Moderate Damage
SafeUnsafe
Example
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2- It is important to define a appropriate thresholds for the probability of failure.
3- The probability of failure computed by means of fragility curve analysis and Monte Carlo analysis shows a maximum difference of 11 % for the case study wall panel. The question is, is this acceptable?
4- In a future study, it could be useful to implement fragility surfaces instead of fragility curves.
5- Also, it could be useful to account for the structural deterioration of the wall panel on computing the fragility curves.
Conclusions
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Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
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Advances in computational methods for resilient structural systemsunder extreme hazards
Minisymposium Organizers: J. Ricles, T. KaravasilisChair: J. Ricles
Pierluigi OlmatiSapienza University of Rome
[email protected] June 12, 2013 -
Fence barrier
Vehicle bomb
w [kgp]
p [W]
Stand-off distance
r [m]
p [R]
Cladding wall
θi
p [Θi]
0
20
40
60
80
100
3.0 3.5 4.0 4.5 5.0P f
(X>
x 0|Z
)
Z
Moderate Damage