FORMAnalysis and SamplingAnalysis

[civilmanjun]

Hi everyone,
In this reliability analysis, FORMAnalysis and SamplingAnalysis are applied, but the results are different evidently, the reliability index of the former is 2.70345, but the latter is 1.1147. The results should be consistent, who can tell me why? thank you


model basic -ndm 2 -ndf 2
reliability

node 1 0 0
node 2 3 0
node 3 6 0
node 4 1.5 3
node 5 4.5 3

set a1 0.00046409
set E 210000000000
set fy 400000000
set p 24123.8250

uniaxialMaterial Steel01 1 $fy $E 0.02

element truss 1 1 2 $a1 1
element truss 2 2 3 $a1 1
element truss 3 1 4 $a1 1
element truss 4 2 4 $a1 1
element truss 5 2 5 $a1 1
element truss 6 3 5 $a1 1
element truss 7 4 5 $a1 1

fix 1 1 1
fix 3 1 1

# LOADS
pattern Plain 1 Linear {
load 4 0 -$p
load 5 0 -$p
}

constraints Plain
numberer RCM
test NormUnbalance 1.0e-6 25 0
integrator LoadControl 0.025 1 0.025 0.025
algorithm Newton
system ProfileSPD
sensitivityIntegrator -static
sensitivityAlgorithm -computeAtEachStep
analysis Static

for { set j 1 } { $j<3 } { incr j 1 } {
randomVariablePositioner $j -createRV3 normal [expr $p*1.06] [expr 0.074*$p] -loadPattern 1 -loadAtNode [expr $j+3] 2}

performanceFunction 1 "0.001255-{u_4_2}"

randomNumberGenerator CStdLib
probabilityTransformation Nataf -print 0
reliabilityConvergenceCheck Standard -e1 1.0e-3 -e2 1.0e-3 -print 1
gFunEvaluator OpenSees -file aux_3.tcl
gradGEvaluator OpenSees
searchDirection iHLRF
meritFunctionCheck AdkZhang -multi 2.0 -add 10.0 -factor 0.5
stepSizeRule Armijo -maxNum 50 -base 0.5 -initial 1.0 2 -print 1 -sphere 1000.0 1.0 1.0
startPoint Mean
findDesignPoint StepSearch -maxNumIter 100 -printDesignPointX designPoint1.txt

runFORMAnalysis Example1a.out
startPoint -file designPoint1.txt
runSamplingAnalysis Example1f.out -type failureProbability -variance 1.0 -maxNum 10000 -targetCOV 0.01 -print 0

[koduru]

I would like to have a look at output of the sampling analysis (output in file Example1f.out).
I am not sure what is the coefficient of variance of the failure probability obtained from sampling.

[civilmanjun]

koduru,
thanks for your help. Here is the

Example1f.out
#######################################################################
# SAMPLING ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Reliability index beta: ............................ 1.1147 #
# Estimated probability of failure pf_sim: ........... 0.13249 #
# Number of simulations: ............................. 10000 #
# Coefficient of variation (of pf): .................. 0.084666 #
# #
#######################################################################


Example1a.out
#######################################################################
# FORM ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Limit-state function value at start point: ......... 5.6546e-005 #
# Limit-state function value at end point: ........... 4.12e-018 #
# Number of steps: ................................... 15 #
# Number of g-function evaluations: .................. 121 #
# Reliability index beta: ............................ 2.7034 #
# FO approx. probability of failure, pf1: ............ 0.0034312 #
# #
# rvtag x* u* alpha gamma delta eta #
# 1 2.557e+004 0.000e+000 0.00000 0.00000 - - #
# 2 3.040e+004 2.703e+000 1.00000 1.00000 - - #
# #
#######################################################################



designPoint1.txt
2.55713e+004
3.03974e+004

[dcizmar]

Hi,

You should take results of FORM (pf) and change one line (i enclose this line and syntax): targetCOV pf

runSamplingAnalysis Example1f.out -type failureProbability -variance 1.0 -maxNum 10000 -targetCOV 0.1963 -print 1

You will get pretty close results...
#######################################################################
# SAMPLING ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Reliability index beta: ............................ 0.84972 #
# Estimated probability of failure pf_sim: ........... 0.19774 #
# Number of simulations: ............................. 40 #
# Coefficient of variation (of pf): .................. 0.18881 #
# #
#######################################################################


Hope this will help...

[civilmanjun]

dcizmar,
thanks for your help, your reply is very useful to me.
but l don't know how to get this data 0.1963 from the results of FORM?

And I input the following file to OpenSees, but get different reliability index, I don't know why.

Example1a.out
#######################################################################
# FORM ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Limit-state function value at start point: ......... 5.6546e-005 #
# Limit-state function value at end point: ........... 4.12e-018 #
# Number of steps: ................................... 15 #
# Number of g-function evaluations: .................. 121 #
# Reliability index beta: ............................ 2.7034 #
# FO approx. probability of failure, pf1: ............ 0.0034312 #
# #
# rvtag x* u* alpha gamma delta eta #
# 1 2.557e+004 0.000e+000 0.00000 0.00000 - - #
# 2 3.040e+004 2.703e+000 1.00000 1.00000 - - #
# #
#######################################################################



Example1f.out#######################################################################
# SAMPLING ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Reliability index beta: ............................ 1.3418 #
# Estimated probability of failure pf_sim: ........... 0.089831 #
# Number of simulations: ............................. 819 #
# Coefficient of variation (of pf): .................. 0.19598 #
# #
#######################################################################



model basic -ndm 2 -ndf 2
reliability

node 1 0 0
node 2 3 0
node 3 6 0
node 4 1.5 3
node 5 4.5 3

set a1 0.00046409
set E 210000000000
set fy 400000000
set p 24123.8250

uniaxialMaterial Steel01 1 $fy $E 0.02

element truss 1 1 2 $a1 1
element truss 2 2 3 $a1 1
element truss 3 1 4 $a1 1
element truss 4 2 4 $a1 1
element truss 5 2 5 $a1 1
element truss 6 3 5 $a1 1
element truss 7 4 5 $a1 1

fix 1 1 1
fix 3 1 1

# LOADS
pattern Plain 1 Linear {
load 4 0 -$p
load 5 0 -$p
}

constraints Plain
numberer RCM
test NormUnbalance 1.0e-6 25 0
integrator LoadControl 0.025 1 0.025 0.025
algorithm Newton
system ProfileSPD
sensitivityIntegrator -static
sensitivityAlgorithm -computeAtEachStep
analysis Static

for { set j 1 } { $j<3 } { incr j 1 } {
randomVariablePositioner $j -createRV3 normal [expr $p*1.06] [expr 0.074*$p] -loadPattern 1 -loadAtNode [expr $j+3] 2}

performanceFunction 1 "0.001255-{u_4_2}"

randomNumberGenerator CStdLib
probabilityTransformation Nataf -print 0
reliabilityConvergenceCheck Standard -e1 1.0e-3 -e2 1.0e-3 -print 1
gFunEvaluator OpenSees -file aux_3.tcl
gradGEvaluator OpenSees
searchDirection iHLRF
meritFunctionCheck AdkZhang -multi 2.0 -add 10.0 -factor 0.5
stepSizeRule Armijo -maxNum 50 -base 0.5 -initial 1.0 2 -print 1 -sphere 1000.0 1.0 1.0
startPoint Mean
findDesignPoint StepSearch -maxNumIter 100 -printDesignPointX designPoint1.txt

runFORMAnalysis Example1a.out
startPoint -file designPoint1.txt
runSamplingAnalysis Example1f.out -type failureProbability -variance 1.0 -maxNum 10000 -targetCOV 0.1963 -print 1

[dcizmar]

Hi, it seems that I misguided you.

Let me explain (i used this last file you posted)

I ran opensees 1.7.3. and get the results:
FORM ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Limit-state function value at start point: ......... 5.6546e-005 #
# Limit-state function value at end point: ........... 1.7347e-018 #
# Number of steps: ................................... 12 #
# Number of g-function evaluations: .................. 79 #
# Reliability index beta: ............................ 0.85491 #
# FO approx. probability of failure, pf1: ............ 0.1963 #
# #
# rv# x* u* alpha gamma delta eta #
# 1 2.702e+004 8.110e-001 0.94868 0.94868 - - #
# 2 2.605e+004 2.703e-001 0.31623 0.31623 - - #
#

SAMPLING ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Reliability index beta: ............................ 0.84972 #
# Estimated probability of failure pf_sim: ........... 0.19774 #
# Number of simulations: ............................. 40 #
# Coefficient of variation (of pf): .................. 0.18881 #
#

From FORM I used pf=0.1963 as COV in simulation.. This, smt. can be used. up there are the results, and I conclueded that this is ok...

I looked Your last post, and figured that we are using different version of OpenSees...

On 1.7.5. version, I also run this example and got this:

#######################################################################
# FORM ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Limit-state function value at start point: ......... 5.6546e-005 #
# Limit-state function value at end point: ........... 4.12e-018 #
# Number of steps: ................................... 15 #
# Number of g-function evaluations: .................. 121 #
# Reliability index beta: ............................ 2.7034 #
# FO approx. probability of failure, pf1: ............ 0.0034312 #
# #
# rvtag x* u* alpha gamma delta eta #
# 1 2.557e+004 0.000e+000 0.00000 0.00000 - - #
# 2 3.040e+004 2.703e+000 1.00000 1.00000 - - #
# #
#######################################################################

#######################################################################
# SAMPLING ANALYSIS RESULTS, LIMIT-STATEFUNCDTION NUMBER 1 #
# #
# Failure did not occur, or zero response! #
# #
#######################################################################

This is very strange. The FORM rel. indexes are very different...Values of function at the end are quite similar..

What version are you running???

[civilmanjun]

dcizmar, I'm so appreciated for your patience and help.

I run it on 1.7.5. version, but I can get the reliability index is 1.3418 in sampling analysis.

[koduru]

Hello civilmanjun,

The primary difference between FORM and sampling results is if sampling has high coefficient of variation (COV). From your results I saw your COV is about 8% which isn't bad.

Therefore, the other causes due to this difference are 1)if the limit-state surface is nonlinear, SORM could give better results. FORM would give erroneous results for highly nonlinear limit-state surface. 2) if your limit-state function/surface have any "kinks" FORM could be converging to a wrong design point.

I suspect your model has both the problems.

Therefore, you could try
1)SORM and compare to sampling
2) change the loading random variables to be a single random variable but with two random variable positioners. (ie create 1 random variable separately and use 2 randomVariablePositioners to point to two nodal loads). Then try comparison of FORM and sampling.

I think the discrepancy between versions could be due to numerical round-off. From the results, the final evaluation of g function is 4.12e-18. Perhaps changing units (from N-m to N-mm) could help in that case.


I hope this helps.

[civilmanjun]

koduru,
I'll try the methods your advised. Thanks a lot.
I am a new user, and your suggestions is valuable to me, thanks again.[/img]

[civilmanjun]

Hi koduru,
FORM, SORM and SAMPLING are applied, and the same results obtained by FORM, SORM, but SAMPLING received different result.
And I decrease the standard deviation of load p, and the coefficient of variance is 0.17%, but similar results are gained as follows:

# FORM
#######################################################################
# FORM ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Limit-state function value at start point: ......... 1.1546e-005 #
# Limit-state function value at end point: ........... 4.3368e-019 #
# Number of steps: ................................... 18 #
# Number of g-function evaluations: .................. 172 #
# Reliability index beta: ............................ 2.9179 #
# FO approx. probability of failure, pf1: ............ 0.0017622 #
# #
# rvtag x* u* alpha gamma delta eta #
# 1 2.557e+004 0.000e+000 0.00000 0.00000 - - #
# 2 2.656e+004 2.918e+000 1.00000 1.00000 - - #
# #
#######################################################################


# SORM#######################################################################
# SORM ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# (Curvatures found from search algorithm.) #
# #
# Number of principal curvatures used: ............... 1 #
# Reliability index beta (impr. Breitung's formula):.. 2.9179 #
# Corresponding estimated probability of failure pf2:..0.0017622 #
# #
#######################################################################



# SAMPLING
#######################################################################
# SAMPLING ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Reliability index beta: ............................ 1.2412 #
# Estimated probability of failure pf_sim: ........... 0.10726 #
# Number of simulations: ............................. 10000 #
# Coefficient of variation (of pf): .................. 0.097398 #
# #
#######################################################################








model basic -ndm 2 -ndf 2
reliability

node 1 0 0
node 2 3 0
node 3 6 0
node 4 1.5 3
node 5 4.5 3

set a1 0.00046409
set E 210000000000
set fy 400000000
set p 24123.8250

uniaxialMaterial Steel01 1 $fy $E 0.02

element truss 1 1 2 $a1 1
element truss 2 2 3 $a1 1
element truss 3 1 4 $a1 1
element truss 4 2 4 $a1 1
element truss 5 2 5 $a1 1
element truss 6 3 5 $a1 1
element truss 7 4 5 $a1 1

fix 1 1 1
fix 3 1 1

# LOADS
pattern Plain 1 Linear {
load 4 0 -$p
load 5 0 -$p
}

constraints Plain
numberer RCM
test NormUnbalance 1.0e-6 25 0
integrator LoadControl 0.025 1 0.025 0.025
algorithm Newton
system ProfileSPD
sensitivityIntegrator -static
sensitivityAlgorithm -computeAtEachStep
analysis Static

for { set j 1 } { $j<3 } { incr j 1 } {
randomVariablePositioner $j -createRV3 normal [expr $p*1.06] [expr 0.014*$p] -loadPattern 1 -loadAtNode [expr $j+3] 2}

performanceFunction 1 "0.001210-{u_4_2}"

randomNumberGenerator CStdLib
probabilityTransformation Nataf -print 0
reliabilityConvergenceCheck Standard -e1 1.0e-3 -e2 1.0e-3 -print 1
gFunEvaluator OpenSees -file aux_3.tcl
gradGEvaluator OpenSees
searchDirection iHLRF
meritFunctionCheck AdkZhang -multi 2.0 -add 10.0 -factor 0.5
stepSizeRule Armijo -maxNum 50 -base 0.5 -initial 1.0 2 -print 1 -sphere 1000.0 1.0 1.0
startPoint Mean
findDesignPoint StepSearch -maxNumIter 100 -printDesignPointX designPoint1.txt
findCurvatures firstPrincipal

runFORMAnalysis Example1a.out
runSORMAnalysis Example1b.out

startPoint -file designPoint1.txt
runSamplingAnalysis Example1f.out -type failureProbability -variance 1.0 -maxNum 10000 -targetCOV 0.0017622 -print 1

[civilmanjun]

hi, everyone

help me, it's a big trouble for me.