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