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One Dimension with Multi MPP Problem
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Description
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Limit State Function
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Random Variable
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Analysis
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Results
This is an a multi MPPs for a given value of g(x) which has
only one random variable. The g(x) is bounded in the range of (0, 1). The result
of one random variable problem by the First-Order Reliability Method (FORM) is
exact solution. UNIPASS has capability to solve one random variable with multi
MPP problem by FORM. Users can use this capability to obtain the PDF/CDF
of any desired variable which is derived from a random variable with then
given statistical information.
X: N(0,0.01)
Perform PDF/CDF analysis with with the range of Pf = (0.001,
0.999) with minimum of 5 points on the curve by FORM and Monte Carlo Simulation Method
FORM with Modified U Based X Linearized MPPL Method
***** SUMMARY OF CDF ANALYSIS *****
First-Order CDF of Limit-State Function 1
Number of points in each CDF curve: 8
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1st-order b
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1st-order Pf
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G-Value = g(x)-g_init
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3.0902340E+00
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9.9999419E-04
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6.0683508E-02
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1.0835546E+00
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1.3928117E-01
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1.3891935E-01
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3.6821173E-01
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3.5635768E-01
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2.1715518E-01
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-1.9670577E-02
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5.0784692E-01
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2.9539102E-01
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-6.7027883E-01
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7.4865997E-01
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5.3009853E-01
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-1.1119938E+00
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8.6692960E-01
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7.6480605E-01
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-1.3722063E+00
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9.1500039E-01
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8.8215980E-01
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-3.0902340E+00
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9.9900001E-01
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9.9951356E-01
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CDF of g by FORM:
***** SUMMARY OF PDF ANALYSIS *****
First-Order PDF of Limit-State Function 1
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1st-order b
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1st-order Pf
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G-Value = g(x)-g_init
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-4.2937854E-05
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1.4457447E-07
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6.0683508E-02
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-1.2604135E+01
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2.7955831E+00
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1.3891935E-01
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-6.5560850E+00
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2.4440711E+00
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2.1715518E-01
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-3.9136221E+00
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1.5610073E+00
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2.9539102E-01
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-2.1482723E+00
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6.8460610E-01
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5.3009853E-01
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-1.3504189E+00
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2.9031485E-01
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7.6480605E-01
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-4.9578016E+00
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7.7147441E-01
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8.8215980E-01
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-9.8489182E-07
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3.3161930E-09
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9.9951356E-01
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PDF of g by FORM:
No. of g-functions called = 240
No. of derivatives of g called = 3
Monte Carlo Simulation
***** SUMMARY OF PDF/CDF RESULTS BY
SIMULATION *****
Based on 50000 Monte Carlo simulations, Limit-State Function: 1
Number of points in each curve: 31
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g(x)-g_init
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PDF of g(x)-g_init
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CDF of g(x)-g_init
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4.4624864E-02
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2.5957870E-03
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3.9999200E-05
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6.7847510E-02
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5.7366894E-01
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8.9198216E-03
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1.0075804E-01
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2.3777409E+00
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5.4398912E-02
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1.3366857E-01
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3.1772433E+00
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1.3999720E-01
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1.6657909E-01
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3.0565392E+00
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2.3605528E-01
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1.9948962E-01
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2.5938402E+00
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3.2312354E-01
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2.3240015E-01
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2.3238784E+00
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3.9890202E-01
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2.6531068E-01
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1.9961602E+00
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4.6547069E-01
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2.9822121E-01
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1.6450800E+00
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5.2157957E-01
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3.3113173E-01
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1.3757671E+00
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5.6812864E-01
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3.6404226E-01
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1.2271583E+00
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6.0823784E-01
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3.9695279E-01
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1.0525916E+00
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6.4336713E-01
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4.2986332E-01
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9.0073810E-01
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6.7346653E-01
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4.6277385E-01
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8.3194975E-01
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7.0016600E-01
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4.9568437E-01
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7.7029981E-01
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7.2485550E-01
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5.2859490E-01
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6.8983041E-01
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7.4735505E-01
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5.6150543E-01
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6.1649942E-01
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7.6748465E-01
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5.9441596E-01
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5.8080735E-01
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7.8593428E-01
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6.2732649E-01
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5.2824266E-01
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8.0302394E-01
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6.6023701E-01
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5.2370004E-01
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8.1923362E-01
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6.9314754E-01
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4.7697587E-01
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8.3465331E-01
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7.2605807E-01
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4.5945431E-01
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8.4908302E-01
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7.5896860E-01
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4.3414538E-01
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8.6285274E-01
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7.9187913E-01
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4.4128380E-01
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8.7634247E-01
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8.2478966E-01
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4.2700697E-01
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8.8972221E-01
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8.5770018E-01
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4.4063485E-01
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9.0309194E-01
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8.9061071E-01
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4.7178430E-01
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1715166E-01
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9.2352124E-01
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5.7237104E-01
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9.3324134E-01
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9.5643177E-01
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6.6062780E-01
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9.5224096E-01
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9.8934230E-01
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1.2193710E+00
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9.8121038E-01
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1.0000000E+00
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1.2193710E+00
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1.0000000E+00
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CDF of g by Monte Carlo Simulation:
PDF of g by Monte Carlo Simulation:
No. of g-functions called = 50000
No. of derivatives of g called = 0
Note: The results from Monte Carlo simulation method verify the result from
FORM.
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