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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: pierluigi.olmati@uniroma1.it

Pierluigi OlmatiSapienza University of Rome

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- June 12, 2013 -

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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- June 12, 2013 -

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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- June 12, 2013 -

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

2

3

4

5

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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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3

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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

1

Pierluigi OlmatiSapienza University of Rome

pierluigi.olmati@uniroma1.itwww.francobontempi.org- June 12, 2013 -

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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- June 12, 2013 -

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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Pierluigi OlmatiSapienza University of Rome

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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

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

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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

2

3

4

5

Intensity measure

Pierluigi OlmatiSapienza University of Rome

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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

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23

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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2

3

4

5

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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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2

3

4

5

SafeUnsafe

Example

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28

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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2

3

4

5

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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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2

3

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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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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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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

pierluigi.olmati@uniroma1.itwww.francobontempi.org- June 12, 2013 -

34

Advances in computational methods for resilient structural systemsunder extreme hazards

Minisymposium Organizers: J. Ricles, T. KaravasilisChair: J. Ricles

Pierluigi OlmatiSapienza University of Rome

pierluigi.olmati@uniroma1.itwww.francobontempi.org- 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