Monte Carlo

Reliability Analysis and Design for Civil Engineers RADCE-09

2009 UVCE AJAY.N

[

RADCE-09, UNDER TEDIP

]

MONTE CARLO SIMULATIONS Dr.V.Devaraj

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2009-10-26

Reliability Analysis and Design for Civil Engineers RADCE-09

MONTE CARLO ANALYSIS

Determine the mean and standard deviation for steel beam, using Monte Carlo simulation method. Data are given below.

SL NO 1 2 3 4 5

VARIABLES fy=250 N/mm2 Z=778.9x103mm3 wg=0.524 N/mm wL=20N/mm

L=6500mm

C.O.V 0.05 0.05 0.10 0.15 0.05

1. CALCULATION OF MEAN VALUES(μ) AND STANDARD DEVIATION "σ":MEAN VALUES=μ=CHARACTERSTIC VALUE1±COFFICIENT OF VARIATION X NORMAL DISTRIBUTION STANDARD DEVIATION=σ=COFFICIENT OF VARIATION X MEAN VALUE

E.g.:- 1)fy=250N/mm2;z=1.645 ; C.O.V=0.05

μfy=fy(1-C.O. V * z)=250(1-0.05* 1.645)=272.4053 N/mm2 σ=C. O. V * μfy=0.05 X 272.4053=13.6202 N/mm2

SL NO 1 2 3 4 5

VARIABLES

Characteristic Value

C.O.V

Mean value(μ)

fy Z wg wL

250 N/mm2

272.40 N/mm2

778.9x103mm3 0.524 N/mm 20N/mm

L

6500mm

0.05 0.05 0.10 0.15 0.05

848.70x103mm3 0.449 N/mm 16.04N/mm

6006mm

Standard deviation (σ) 13.62 42.43 x103 0.0449 2.406 300.3

2. STRENGTH FORMULATION :

Dr.V.Devaraj

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2009-10-26

Reliability Analysis and Design for Civil Engineers RADCE-09

MS =R-S

Note: - For calculation purpose, considering the mean values. MS =fY*Z- Wg +WL * L28

Generate 10 random numbers

Normal Distribution:

Dr.V.Devaraj

SL NO

Random number

1

52478

2

80249

3

94132

4

56605

5

58815

6

69379

7

75228

8

14327

9

90625

10

06070

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Reliability Analysis and Design for Civil Engineers RADCE-09

Transformation

x=μ+σ ∅-1(Pi)

1. μfy=272.40 N/mm2 And σfy=13.62N/mm2 SL NO

Random number

1 2 3 4 5 6 7 8 9 10

52478 80249 94132 56605 58815 69379 75228 14327 90625 06070

From Distribution Table 0.0622 0.8507 1.5658 0.1662 0.2225 0.5065 0.6818 -1.0656 1.3182 -1.549 z=∅-1(Pi)

Pi

0.52478 0.80249 0.94132 0.56605 0.58815 0.69379 0.75228 0.14327 0.90625 0.06070

x

273.24 283.98 293.72 274.66 275.43 279.29 281.68 257.88 290.35 251.30

2. μz=848.70x103mm3 And σz=42.43 x103mm3.

Dr.V.Devaraj

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2009-10-26

Reliability Analysis and Design for Civil Engineers RADCE-09

∅-1(Pi) From Distribution Table

x

0.52478

0.0622

851.33e3

80249

0.80249

0.8507

884.79e3

3

94132

0.94132

1.5658

915.13e3

4

56605

0.56605

0.1662

855.75e3

5

58815

0.58815

0.2225

858.14e3

6

69379

0.69379

0.5065

870.19e3

7

75228

0.75228

0.6818

877.62e3

8

14327

0.14327

-1.0656

803.48e3

9

90625

0.90625

1.3182

904.63e3

10

06070

0.06070

-1.549

782.97e3

SL NO

Random number

1

52478

2

Pi

3. μwg=0.449 N/mm And σwg=0.0449 N/mm SL NO

Random number

1 2 3 4 5 6 7 8 9 10

52478 80249 94132 56605 58815 69379 75228 14327 90625 06070

Pi

0.52478 0.80249 0.94132 0.56605 0.58815 0.69379 0.75228 0.14327 0.90625 0.06070

From Distribution Table 0.0622 0.8507 1.5658 0.1662 0.2225 0.5065 0.6818 -1.0656 1.3182 -1.549 ∅-1(Pi)

x

0.501 0.537 0.569 0.506 0.508 0.521 0.529 0.451 0.558 0.0429

4. μwL=16.04 N/mm And σwL=2.406 N/mm

Dr.V.Devaraj

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2009-10-26

Reliability Analysis and Design for Civil Engineers RADCE-09

SL NO

Random number

1 2 3 4 5 6 7 8 9 10

52478 80249 94132 56605 58815 69379 75228 14327 90625 06070

Pi

0.52478 0.80249 0.94132 0.56605 0.58815 0.69379 0.75228 0.14327 0.90625 0.06070

From Distribution Table 0.0622 0.8507 1.5658 0.1662 0.2225 0.5065 0.6818 -1.0656 1.3182 -1.549 ∅-1(Pi)

x

16.18 18.08 19.80 16.43 16.57 17.25 17.68 13.47 19.21 12.31

5. μL=6006mm And σL=300.3mm SL NO

Random number

1 2 3 4 5 6 7 8 9 10

52478 80249 94132 56605 58815 69379 75228 14327 90625 06070

Pi

0.52478 0.80249 0.94132 0.56605 0.58815 0.69379 0.75228 0.14327 0.90625 0.06070

∅-1(Pi) From Distribution Table 0.0622 0.8507 1.5658 0.1662 0.2225 0.5065 0.6818 -1.0656 1.3182 -1.549

x

6024.67 6260.61 6475.20 6055.90 6072.81 6158.10 6210.74 5685.00 6401.85 5540.83

3. CALCULATION OF MARGINAL SAFETY FACTOR: MS=μfy*μZ-(μwg+μwL)8*μL2

MS=μR-μS MS=R-S

Calculation of μR and σR:

Dr.V.Devaraj

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Reliability Analysis and Design for Civil Engineers RADCE-09

μR=μfy*μZ

Me an= μR=238.89 x106 ;

1

273.24*851.33=232.61e6

2

251.26e6

σR=X-X2n-1=22.45 x106

3

268.79e6

4

235.04e6

5

236.35e6

6

243.03e6

7

247.20e6 μS=(μwg+μwL)8*μL2

18 29

207.20e6 75.68e6 262.65e6 91.21e6

10 3

196.76e6 106.75e6

Calculation and σS:

of

μS

4

77.63e6

5

78.72e6

6

84.23e6

7

87.79e6

8

56.23e6

9

101.27e6

10

47.40e6

Mean=μS=80.68 x106 ; σS=X-X2n-1=18.32x106 4. CALCULATION OF RELIABILITY INDEX “β": β= (μR-μS)σR2+σS2 β= (238.89 x106-80.68 x106)(22.45x106) 2+(18.32x106)2=5.45 COVR= σRμR=22.45 x106238.89 x106=0.09 COVS=σSμS=18.32x10680.68 x106=0.22

5. CALCULATION OF CENTRAL SAFTY FACTOR: Central Safety Factor =RS=rϕ ϕ=1-0.75β COVR=1-0.75 X 5.45 X 0.09=0.63 γ=1+0.75β COVS=1+0.75 X 5.45 X 0.22=1.89

Dr.V.Devaraj

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2009-10-26

Reliability Analysis and Design for Civil Engineers RADCE-09 Central Safety Factor =238.89 x10680.68 x106=1.890.63=3.0

It satisfy the condition,

ϕR≥γS 0.63*238.89 x106≤1.89*80.68 x106 150.50x106≤152.48 x106

7. REFERENCES [3.1] RAND Corporation (1955): A Million Random Digits with 100,000 Normal Deviates, the Free Press, and Glencoe, IL. [3.2] RUBENSTEIN, M.F., and R.Rosen (1968):”Structural Analysis by Matrix Decomposition,” Journal of the Franklin Institute, Vol.286, No.4. Additional Reading BENJAMIN, J.R., and C.A. CORNELL (1970): Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill Book Company, New York. HAUGEN, E.B. (1968): Probabilistic Approaches to Design, John Wiley & Sons, Inc., New York. RUBINSTEIN, M.F. (1975): Patterns of Problem Solving, Prentice-Hall, Inc Englewood Cliffs, NJ. GORDON, G., and I. PRESSMAN (1978): Quantitative Decision-Making for Business, Prentice-Hall, Inc., Englewood Cliffs, and NJ.

Dr.V.Devaraj

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2009-10-26