Harnessing the Use of Ordinary Monte Carlo Simulation

7/31/2019 Harnessing the Use of Ordinary Monte Carlo Simulation

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InCiTe 2012 (BAYLON &GARCIANO)

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8/20/12 inCiTe 2012 (BAYLON & GARCIANO)

Harnessing Ordinary Monte Carlo Simulation inComputing the Performance of a TransportationLifeline in the Philippines, the Light Rail TransitSystem under a large magnitude earthquake

Michael B. BAYLONInstructorFar Eastern University East Asia College

Lessandro Estelito O. GARCIANOAssociate ProfessorDe La Salle University


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8/20/12 InCiTe 2012 (BAYLON & GARCIANO)

Introduction

Assessment

Mass Railway Transit

Serviceability

Retrofitting


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Specific Objectives


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Objective

Assessment of a Transportation Lifelinein Manila

Reliability IndexOrdinary Monte Carlo Simulation


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Technology

Software:

MatLab Script for Newmark Method

MatLab Script for Ordinary MonteCarlo Simulation

Hardware:

Reinforced Concrete ColumnConfinement Model


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methodology

Preliminaries & Case Study


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Flow of Methodology


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Contents

Preliminaries

Case Study


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Preliminaries


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Case Study


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Performance Function, g

a.k.a. limit state function

where:

R = resistance or capacity

S = load or demand


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Probability of Failure


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Sample of R vs. S plot


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Probability of Failure

Alternatively,


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Reliability Index

Hasofer-Lind Reliability Index of 1974

or


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Limit State Function (2D)


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Crushing Failure vs BucklingFailure

I CiT 2012 (BAYLON &


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Confinement Model



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Case study



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Structural Plans

Figure 2. A Typical Elevation View of LRT 1 Pier



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Structural Plans

Figure3. A Typical Section View of LRT 1 Pier



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General Notes

InCiTe 2012 (BAYLON &


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Resistance, R

a.k.a. capacity

Ultimate Axial Strength, Pu



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Resistance

Property Mean value coeff.of

variation

standard

deviation

Concrete compressive strength

fc = 3 ksi 2.760 ksi 0.18

fc = 4 ksi 3.390 ksi 0.18

fc' = 5 ksi 4.028 ksi 0.15

1 ksi = 6900 Pa; 1 in = 25.4 mm

A range of values is presented in some instances because data from multiple sourceswere used.

Source: Ellingwood, Galambos, MacGregor, and Cornell, 1980.

Statistical parameters of material properties and dimensions



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Buckling



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Confined vs. Unconfined

Stress-Strain Diagram of Confined and Unconfined Concrete Column. (Source: Miller, 2006)



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Load, S

1 Dead Load + 2 Seismic Load

Dead Load = self-weight of the pier

Deterministic

Seismic Load (Probabilistic)

Level 1: Imperial Valley Eq (El Centro of 1940)

Level 2: Tohoku-Kanto Eq of March 2011.

Mean & Std. Deviation of Spectral Acceleration byNewmark Beta Method



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Newmark Beta Method



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Table3. Summary of parameters of variables

1Using Newmark Beta Method for Tohoku-Kanto Earthquake2Using Newmark Beta Method for El Centro Earthquake

Parameters of Variables

Variable Distribution Mean Standard

Deviation

Coefficient

of Variance

RESISTANCE

fco Normal 23.734 MPa --- 0.18

fyh Normal 312.33 MPa 36.542 MPa 0.116

s Normal 1.171E-03 1E-06 ---

LOAD

1PEQ Lognormal 4580 kN 4001 kN ---

2PEQ Lognormal 2442 kN 2440 kN ---



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Performance Function, g



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Ordinary Monte CarloSimulation



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RESULTS



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Simulation Results (1E5 data points)

Unconfined Confined % diff Unconfined Confined % diff

Tohoku-Kanto Earthquake ( M9.0) El Centro

Pf 0.00122 0.00045 63% 0.00026 0.00005 81%

3.03 3.32 10% 3.47 3.89 12%

Mean s, mm - - - 300

- - - 299.9

Mean fco, MPa 23.74 23.76 23.73 23.75

Mean fyh, MPa 312.3 312.4 312.2 312.4

Mean rho 0.017107 0.017107 0.017107 0.017107

Q, Newtons 6097894 6092409 3450067 3459735



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Load vs. Resistance of Unconfined Column(Tohoku - Kanto EQ), Pf=0.001220, =3.03

0 0 0 0 0 00 00

x 000

0

0

0

0

0

00

00x 00

0

Resistance

Load



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Load vs. Resistance of Confined Column(Tohoku - Kanto EQ), Pf=0.000450, =3.32

0 0 0 0 0 00 00

x 000

0

0

0

0

0

00

00

00x 00

0

Resistance

Load



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Load vs. Resistance of Unconfined Column(El Centro EQ), Pf=0.000260, =3.47

0 0 0 0 0 00 00

x 000

0

0

0

0

0

00

00x 00

0

Resistance

Load



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Load vs. Resistance of Confined Column(El Centro EQ), Pf=0.000050, =3.89

0 0 0 0 0 00 00

x 000

0

0

0

0

0

00

00x 00

0

Resistance

Load



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Specification of Machine Used inMCS & NBM Implementation



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ANALYSIS



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ANALYSIS

there is a significant decrease ofPf

From 122% to 0.045%,

per cent difference of 63%,

Introduction of confinement model to the RC Pier

Tohoku-Kanto Earthquake simulation

For El Cenro Earthquake simulation:

higher per cent difference of 81%



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ANALYSIS

In the ordinary MCS of the Tohoku-KantoEarthquake,

Resistance range of approx. 2x107 N to

8x107 N has changed to a range of 3x107N to 9x107 N

Due to the change RC column

configuration, i.e. confinement.

Approx. same with El Centro



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ANALYSIS

In terms of the load range

from approx. zero magnitude

As high as 11x107 N on both earthquakesimulations.



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ANALYSIS

For the software side

ordinary MCS and Newmark Beta Method (NBM)

Made possible and relatively easier to compute

MatLab built-in functions

Inherent matrix or vector manipulation

Specifications of the machine used for fast computing.



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ANALYSIS

MatLab Built-in Function/Syntax

Description Task

normrnd(mu,sigma) generates random numbers from the

normal distribution with mean

parameter and standard deviation

parameter .

Used in simulation of values for

parameters, e.g. tie spacing, specified

concrete strength, steel yield strength,

steel ratio

lognrnd(mu,sigma) returns an array of random numbers

generated from the lognormal

distribution with parameters and .

Used in simulation of values for load

function, e.g. Tohoku-Kanto Earthquake

induced inertial force



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ANALYSIS

MatLab Built-in Function/Syntax

Description Task

norminv(P,mu,sigma) computes the inverse of the normal cdf

using the corresponding mean and

standard deviation at the

corresponding probabilities in P.

Used in computing the Holzen-Lind

reliability index, , from the computed

probability of failure, Pf.

plot(X1,Y1,...,Xn,Yn) plots each vector Yn versus vector Xn

on the same axes.

Used in plotting the Load-Resistance

relationship which is important in

determining the Failure and Safe zonesby visualization.



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Discussion

Conclusion & Recommendations



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CONCLUSIONS

The reliability indices:

3.03 (unconfined) and 3.32 (confined)

simulated under a Tohoku-Kanto Earthquake.

3.47 (unconfined) and 3.89 (confined).

the simulation of El Centro Earthquake

effectiveness of the confinement model used in

this simulation

average of 11.5% improvement of confinement in thereinforced concrete pier.



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CONCLUSIONS

Efficient use of MatLab in carrying out theimplementation of MCS and Newmark BetaMethod.

MatLabs built-in functions generating random values for the simulation process

Inherent matrix and vector operations

Produce 2D plot of important Load-Resistancerelationship

The capacity of computer used in the source coderuns while using the software MatLab

number of iterations reached an order of 5.



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CONCLUSIONS

Based from the ordinary Monte CarloSimulation, the light railway transit, in itsmaiden structural form, can withstand

seismic forces. IT IS SAFE.



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RECOMMENDATIONS

A more sophisticated method ofstructural reliability is suggested toobtain the value of reliability index or

probability of failure of the structure



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RECOMMENDATIONS

A series of known ground motion data,specifically ground acceleration whichtaking into account the soil type, must

be used to compute for the reliabilityindex.



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RECOMMENDATIONS

Instead of a simple SDOF lumpedmass model, structural modeling thruthe use of finite element methods must

be used for an accurate account of thephysical properties of one of the LRTsreinforced concrete pier.



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RECOMMENDATIONS

Since this research dealt only one partof the structural system, i.e. column, amore detailed structural reliability study

of the system can be implementedusing the as-built plans of a certain lineof the LRT System.



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REFERENCES



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[9] MICHAEL, A.P., HAMILTON, H.R., and ANSLEY, M.H., Concrete Confinement Using

Carbon Fiber Reinforced Polymer Grid, pp 991 1009.



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[11] YAMAZAKI, F. and SHINOZUKA, M., Safety Analysis of Stochastic Finite Element Systems ByMonte Carlo Simulation, Proceedings of Japan Society of Civil Engineers, Structural Engineering /Earthquake Engineering, Vol. 5, No. 2, October 1988, pp 313-323.

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[14] FRANGOPOL, D.M., IDE, Y., SPACONE, E., and IWAKI, I., A New Look At Reliability ofReinforced Concrete Columns, Structural Safety, Vol. 18, No.2/3, Elsevier Sciece Ltd., (c) 1996, pp123 - 150

[15] MITROPOULOU, CH.CH., LAGAROS, N.D., and PAPADRAKAKIS, M., Life-Cycle CostAssessment of Optimally Designed Reinforced Concrete Buildings Under Seismic Actions, ReliabilityEngineering and System Safety, 2011, pp 1311-1331.

[16] TORREGOSA, R., SUGITO, M., and NOJIMA, N., Assessment of Seismic Hazard andMicrozoning in the Philippines, Journal of Structural Mechanics and Earthquake Engineering,

[17] RIEDERER, K.A., Assessment of Confinement Models For Reinforced Concrete ColumnsSubjected To Seismic Loading, Masters Thesis, University of British Columbia, (c) 2006.

[18] DER KIUREGHIAN, A., HAUKAAS, T., and FUJIMURA, K., Structural Reliability Software At TheUniversity of California, Berkeley, Structural Safety 28, (c) 2006, pp 44-67.

[19] GARCIANO, L. E. and YOSHIDA, I., Reliability analysis of a brittle fracture due to crackinstability using sequential Monte Carlo simulation, Proceedings of the Internationl

Conference on the Applications of Statistics and Probability, pp 2949 -2956, 2011.



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Harnessing the Use of Ordinary Monte Carlo Simulation