Multipurpose Probabilistic Software for Statistical
Contents
1. jk General Data Check samples a 20 Bl Model Analysis 1 Je Simulation Resuts Assesene Requested 4 Obtsined 11 9 0 06 0 04 0 02 10 20 30 40 T Numbers 2 c C PoF 5 All iCaiesenDnodnaled C CDF Notam El Load Resistance Comparative values All variables i Load a Load F Resistance Ast Resistance b Resistance Rsd Resistance Rbd E Load a 1 0 n 0 0 Load F 0 0014146 1 0 0 0 0 Resistance Ast o 0 1 a 0 0 i 21 Resistance 0 0015417 00015573 o 1 0 0 Figure 17 Check samples window sampled variable 15 6 3 Model analysis 6 LATIN HYPERCUBE SAMPLING AE File Edit Mew Help Ca S E M Stochastic model X2 vs X1 La Random variables Correlation melet Fequestec 0 9 Obtained 0 674 Eror 0 0262 de Statisti EK Latin Hyper Sampl AEH General Data Check samples E Model Analysis Le Simulation Results Assessment x 0 75 0 874 0 6 0 748 0 6C3 1 0 002 0 00126 0 000716 0 0111 0 468 0 00114 0 000358 7 94e 005 0 000621 0 00105 0 298 0 000753 Category 1 Comparative values All variables Figure 18 Check samples window sampled correlation 6 3 Model analysis Repetitive calculation of response limit state functions is started when the user activates this button Repetitive calculation is performed and r2. saved Responses associated with monitoring points are transferred into FReET after simulation process is finished 16 FREET PROGRAM DOCUMENTATION USER MANUAL ozi Ele Edt Wew Help O SHAE efi Stochastic model La Random variables Statistical correlation B BC Latin Hypercube Sampling of General Data Check samples E Model Analysis x Ela Simulation Results Assessme EX Histograms gt ol y Start Model Analysis YE Sensitivity analysis la Feliabiity New Model Function Delete Model Function Run Model Analysis E of the DLL Exported functions Result name i aluator on ath x3 x5 x4x6 a xu pes mmm Figure 19 Input of model functions window 7 SIMULATION RESULTS ASSESSMENT 7 1 Histograms After simulation process successfully finishes the resulting set of values of response limite state functions e g structural responses can be statistically evaluated The results are histogram emirical cumulative distribution function mean value variance coefficient of skewness min and max values range response This basic statistical assessment is visualized through the window Histograms Fig 20 7 2 Limit state function LSF definition The window is designed for definition of a limit state function by combination of a response variable obtained via simulation and a comparative value Usual type of combination is in the form Z
3. Category 3 etc This order should correspond with a vector of random variables defined in special DLL unit written in C FORTRAN or other programming languages The structure of a special DLL unit of M version is described in chapter 5 1 In case that only a simple function is treated using equation interpreter chapter 5 1 the list of variables names and related categories will appear in equation interpreter window A version Input parameters of ATENA deterministic computational model are trans ferred into FReET name and deterministic value Groups in FReET are equivalent to numbers of materials defined in ATENA Number of transferred random variables of the model both material and geometrical can be initially filtered decreased at the level FREET PROGRAM DOCUMENTATION USER MANUAL Distribution details B xj Distribution Status O D K Normal Moments Parameters gt r Moments amp params the Hem 25 E 25 Mean 25 Median 25 p s j2 sid 25 Mode 25 l Std 125 Std Cov foi Skew B Kurt Ro Calculator X a POF x 3 07788 023 CDF 7 6199e 024 INWip 25 Figure 8 Distribution support calculation E m Characteristics Concrete 1 1 General al I Distri Deterministic PGR 111 Concrete JESS F Mean 303 PEP Ready mixed 015 MW Std 4 26 ompre ength Tensile strength HFPa Modulu
4. E un j Ready Figure 15 Imposing of statistical correlation 6 2 Check samples The aim of this entry is to have the possibility to check the results of sampling scheme applied to random variables before running repetitive very often time demanding cal culation Achieved correlation matrix after simulated annealing is visualized in lower 14 FREET PROGRAM DOCUMENTATION USER MANUAL Reached correlation Maximum deviation in correlation matris 0 000275 Al pt bn LIU SUL UY eal rut Figure 16 Information about achieved accuracy triangular part of correlation matrix upper triangle contains desired target correlation If user clicks on diagonal upper part window will show associated sampled variable Fig 17 If the click is targeted to correlation coefficient out of diagonal the image of sampled values is shown where correlation is clearly visible Fig 18 Cartesian or parallel coordi nates can be selected Before model analysis is performed the option g X lt 0 is not active Not drawn is shown After performing model analysis a particular limit state function can be selected here Then generated points located in safe region are shown by green dots in failure region by red dots Cartesian coordinates plot EE a Edit wiew Help DS tel ARE AP Stochastic model a Random variables la Distribution Normal HK Statistical correlation Mean Latin Hypercube Sampling Requested 20
5. R E 2 where R is a resistance and E action of loading carrying capacity limit state For serviceability limit state the form is just opposite comparative value response e g allowable maximum deflection real deflection The concept of all possibilities of combinations of response monitor variables and com parative values uses basic algebraic operations to define any basic form of limit state function Standard classical picture comparing histograms of R and F is illustratively shown including safety margin Z Fig 21 17 7 3 Sensitivity analysis 7 SIMULATION RESULTS ASSESSMENT T Benchmarlc4 fre Freet ID xi File Edit Mew Help eee ei ETM Stochastic model Xlimit 1 La Random variables ARG Statistical correlation Ex Latin Hypercube Sampling 1 Std AEH General Data Check samples E Model Analysis l Elas Simulation Results Assessment lan Histograms Mean 1 R E LSF definition mz Sensitivity analysis WA Reliability Digits Drawing lt A e por C cor Result mame Classes Warlarice Skewness Xlimit 1 10 004e 00 0 1282 Figure 20 Basic statistical assessment histogram and statistical characteristics i Benchmark4 fre Freet Ble Edt Mew Help AA E Stochastic model La Random variables E Statistical correlation E Latin Hypercube Sampling EH General Data le Check samples EN Model Ana
6. e name Dio now Digits Database RawDats Details F El E POF CDF Name Distribution Descriptors Mean Std cov Skewmess Kurtosis Status 1 E Weibullmax EVI Y Moments 30 3 0 1 116182011 0 28314 OK normal lomera NN KS NTI n ji aE Eine Ziomens 2 25 84 a a OK concrete Comparative values Ready C m 4 Figure 4 Probability distribution window PDF Freet Freet AE Bile Edit wew Help De a eS E TM Stochastic model Random variables iG Statistical correlation r Latin Hypercube Sampling Le Simulation Results Assessment New Name Delete New Delete Database Raw Data Details 5 POF COR Mame Distribution Descriptors Mean Std cov Skewness Kurtosis Status 30 O1 116188011 0 28314 O K Category e support calculation E o il E WeibullmaxEv IIT y Moments 2 fe Normal z Moments 5 01 0 n OK concrete Comparative values Ready um A Figure 5 Cumulative probability distribution window CPDF But this category is not always utilized in case we analyze only a response function or all variables of limit state function are defined directly only using equation interpreter or DLL function section 5 1 Note that in case we achieved results through FReET usage analysis is performed and Simulation results as
7. of random variables which exhibits statistical correlation as close as possible to prescribed correlation matrix Note that in case of very small number of simulations e g tens imposition of prescribed correlation is difficult and maximum deviation in Reached correlation window can be high The window contain also the possibility to select seed setting for pseudo random gen erator used Random means random setting based on internal clock of computer the generated series will differ if used second time Fixed means that the same seed is se lected it enables to generate same series Fixed setting can be efficiently used during debugging of analyzed task AE Ble Edit Mew Help EEE Stochastic model Correlation control norm vs switches lan Random variables HE Statistical correlation 18 Ela Latin Hypercube Sampling M General Dala Check samples 12 E Model Analysis Actual sample Elle Simulation Results Assessme 1 1 7685e 005 Best sample 0 8 1 76858 005 Temperature DE 1 5967e 008 Number of simulations Simulated annealing M Default parameters Sampling ype 77 Random sampling first p Automatic stop LHS median No ofleope jeg ESTO LHS mean And Or LHS random PAE F Ici Cii lt nat Monte Carlo Loops per T Ji50 p Estimated lime pe p W Importance sampling Ties 1 67 Random Tita EE Fed at a pil
8. the value Optionally statistical characteris tics or statistical parameters or combination of characteristics and parameters are used to describe distribution menu Descriptors Parameters symbols are unique for each dis tribution and the meaning of parameters is explained fully in theory guide of FReET The shape of probability distribution of particular random variable is shown in main graphical window checkbox Drawing serves for selection of probability distribution PDF Fig 4 or cumulative probability distribution CPDF Fig 5 windows Random variables can be divided into several categories see bottom of the window User can select a new category and within a selected category a new variable This option is included in order to make handling of large number of random variables easier and more transparent The result of this step is the set of defined input parameters for computational model random variables The category Comparative values is always included in window and can be used in the limit state function definition Simulation Results Assessment LSF definition FREET PROGRAM DOCUMENTATION USER MANUAL Freet Freet it view Heip Ce Bee S ze E TM Stochastic model La Random variables Statistical cunelalivn E Latin Hypercube Sampling La Simulation Results Assessment Category Variable Distribution support calculation Drawing
9. ET is provided in FReET Part 2 Theory FREET PROGRAM DOCUMENTATION USER MANUAL 2 FILE 2 1 Open a FReET data file File Open A FReET file can be opened by this command A classical window shown in Fig 1 will appear after using this command FReET files have extension fre and contain input data and results Ce AX Oblast hled n Sy Benchmark es Ea N zev souboru Soubory typu Frest Files fre prb Storia ll Figure 1 Open existing FReET file Input fre files with benchmarks can be found in subdirectory Examples of the program FReET in case of the default setting is used during installation of the program History of recently opened files is saved during one activation of FReET under File menu for easy handling Note In case that FReET HASP hardlock is not found the program is functioning as a demo version with several restrictions which is announced One restriction is that only rectangular distribution is allowed for basic random variables In case fre file is opened under running demo version all distributions are changed to rectangular ones 2 2 Saving a file File Save Save as A current FReET file can be saved using this commands A dialog window appears if the file has not yet been given name or if Save as command was clicked Fig 2 A file name should be written in this dialogue box and by clicking Save button the file is saved 2 3 Ending FReET File Ex
10. Lo Reliability I Numbers C Paralel Coordinates Cartesian Coordinates Name sensiy sensi x 1 Mame sensiv7 sensi x 1 5 Category 1 45 z T O al 6 Category 1 X6 0 508 ml 2 Limit State Function 1 0 12 3 Category 1 43 0 197 M 2 Category 1 42 0171 k Category 1 X1 0 122 TF 4 Category 1x4 0 073 ly Ready mla Figure 24 Window Sensitivity Analysis positive sensitivity cartesian coordinates 20 FREET PROGRAM DOCUMENTATION USER MANUAL 7 Benchmarl4 fre Freet Ex File Edt Mew Help SH teea El TM Stochastic model La Random variables HE Statistical correlation EX Latin Hypercube Sampling F General Data 4 Check samples JB Model Analysis Elas Simulation Results Assessment in Histograms R E LSF defintion mz Sensitivity analysis W Reliability I Numbers Parallel Coordinates Cartesian Coordinates Name sensi sensi Mame sensi sensi Category 1 05 o7 JI E R 0 741 Category 1 X6 0 508 Limit State Function 1 0 12 Category 1 X3 Category 1 X2 Category 1 41 Category 1 X4 Figure 25 Window Sensitivity Analysis negative sensitivity cartesian coordinates gt Benchmarkfre Freet lnx Bile Edit Mew Help Dem mes m El TM Stochastic model La Random varia
11. Multipurpose Probabilistic Software for Statistical Sensitivity and Reliability Analysis T MM 2 _ a KI PROGRAM DOCUMENTATION Revision 6 2006 FReET version 1 4 Part 2 FReET M A User Manual Cervenka Consulting Prof Drahom r Nov k Ph D Saumannova 10 615 00 Brno Czech Republic phone 420603172861 e mail novak d karneval cz University address Institute of Structural Mechanics Faculty of Civil Engineering Brno University of Technology Veveri 95 602 00 Brno Czech Republic Distributor Vladim r ervenka Ph D Cervenka Consulting Predvoje 22 162 00 Prague 6 Czech Republic phone fax 420220610018 e mail cervenka cervenka cz http www cervenka cz FReET Program Documentation Part 2 User Manual Written by Drahom r Nov k Radoslav Rusina Miroslav Vo echovsk Brno May 2006 FREET PROGRAM DOCUMENTATION USER MANUAL Contents 1 INTRODUCTION 2 2 FILE 3 2 1 Open a FReET data file File Open lt 444 3 2 2 Saving a file File Save Save as 4 skor Sogo Poe k jel oby hou 3 2 3 Ending FReET File Exit aig A Vo a 3 3 MAIN PROGRAM TREE 3 4 STOCHASTIC MODEL 5 4 1 Random variables ooa 2 ee ee 5 4 2 Statistical correlation lt lt lt lt lt lt 4 4 e 10 5 RESPONSE LIMIT STATE FUNCTION DEFINITION 11 del EREE MEVETSIOOA a dogs ee ee gS AAA ee ee EEE a 11 5 1 1 Equation interpreter lt 444 11 5 1 2 DEL FUNGO teta ele al
12. bel min EV I 4 1 Random variables 4 STOCHASTIC MODEL e Gumbel max EV I e Rectangular e Triangular e Laplace e Pareto e Logistic e Half Normal e Half Normal negative e Beta e Student t Note that some distributions can be defined by a certain setting of parameters of other distributions The simpler forms are included in the list to allow easy handling by user Details on probability distributions are in theory guide of FReET Random variable can be described also by raw data here select Distribution support calculation and Raw Data Fig 7 In this case the name of input text file with statistical set arranged in columns or rows in ASCII format is reguired or data can be directly written into the edit box Then selection Calculate parameters will cause also curve fitting the selection of most suitable probability distribution is performed distributions are ordered according to the confidence levels SL of curve fitting By clicking Apply the result of raw data assessment is transferred into variable definition window Random variables should be basically described by statistical characteristics statistical moments Mean value standard deviation or coefficient of variation and coefficient of skewness respectively Exceptionally also 4th statistical moment kurtosis is used Beta distribution Standard deviation or coefficient of variation is recalculated automatically with respect to mean value when entering
13. ble REliability Engineering Tool FReET can be utilized in two versions as stand alone multipurpose program for any user defined problem M version and as module integrated with ATENA A version This manual provides the descriptions of general features of the software valid for both ver sions Main differences between two versions are described in the chapter Response Limit state function definition If necessary the differences are mentioned as a remark related to M or A versions in other chapters of this manual The main aim of this text is to describe how to efficiently utilize software FReET with all details and possibilities provided by the graphic user interface FReET A version has been integrated with advanced nonlinear fracture mechanics software for computational analysis of concrete structures the finite element program ATENA Cervenka Consulting Prague Czech Republic the integration is controlled by SARA Studio software shell Full understanding of the concept of this complex integration is beyond the framework of this text The user of A version of FReET should get next information support from ATENA and SARA documentations Although some examples are included in this manual to support the description of some software functions a systematic treatment of examples is not covered It is a subject of a separate document FReET Part 3 Benchmarks and Examples The description of theoretical methods implemented in FRe
14. bles HE Statistical conelation Et Latin Hypercube Sampling H General Data Check samples JB Model Analysis Ela Simulation Results Assessment in Histograms R E LSF definition Sensitivity analysis La Reliability Parallel Coordinates Cartesian Coordinates I Numbers Name sensi sensi x l Mame sensiiy sensi Category 1x1 1 Fr R Loi m 0 41 a Limit State Function 1 Figure 26 What if study using sensitivity analysis window 7 4 Reliability analysis Histogram of response and or safety margin as specified in limit state function definition is visualized The aim of this window is to provide an estimation of theoretical failure probability and reliability index respectively Fig 27 Theoretical models of PDF in 21 7 4 Reliability analysis 7 SIMULATION RESULTS ASSESSMENT order to describe the most suitable models are treated using a standard Kolmogorov Smirnov test curve fitting CF procedure The most suitable model is listed in CF Distribution at the top according to the significance level CF SL Distribution details button enables the same features as in case of basic random variables input in Stochastic model For example confidence intervals for results can be easily determined utilizing the possibility Following alternatives are implemented Cornell s rel
15. dow Edit response limit state function 5 1 2 DLL function The analyzed response limit state function is defined completely outside as subroutine written in C FORTRAN or other programming languages This subroutine has to be compiled to DLL function FReET response limit state function is then implemented into FReET as DLL unit The structure of an DLL program unit should follow prescribed convention We provide here self explanatory example for simple function B GX 4 5 1 Program units in C and Fortran are shown in Fig 13 and Fig 14 Note that A input 0 B input 1 C input 2 in C and A input 1 B input 2 C input 3 in Fortran Note that the number of random variables in DLL function should correspond with number of random variables defined in Stochastic model This is fully the responsibility of an user FReET cannot check this fundamental requirement finclude math kh _declspec dllesport double stdoall Xlimitt double input double ret U ret input U input 1 input 2 return ret Figure 13 The structure of external program unit in C 12 FREET PROGRAM DOCUMENTATION USER MANUAL real s function KlimitF input IDECS ATTRIBUTES DLLEEPORT ElimitF realx 8 input HlimitF 0 KlimitF input li input 24i input 3 end function ElimitF Figure 14 The structure of external program unit in Fortran 5 2 FReET A version ATENA computational model of nonlinear fracture
16. ean Std COW Cornell B Cornell pF CF Distribution CF SL MFiMtot COV pF Xlimit 1 10 270 0 392 2 62 0 00439 Weibull max 3 par 7 0 00849 0 727 0 01 Figure 27 Reliability analysis window 22 FREET PROGRAM DOCUMENTATION USER MANUAL References 1 ervenka V amp Pukl R 2003 ATENA Program Documentation Theory Cer venka Consulting Prague Czech Republic ervenka V amp Pukl R 2003 ATENA Program Documentation User s guide Cervenka Consulting Prague Czech Republic Nov k D Vo echovsk M Tepl B Ker ner Z amp Lehk D 2004 FReET Program Documentation Part 1 Theory Brno Cervenka Consulting Prague Czech Republic 4 Novak D amp Lehk D 2004 FReET Program Documentation Part 3 Bench marks Brno Cervenka Consulting Prague Czech Republic 5 ervenka V amp Pukl R 2004 SARA Program Documentation User s guide Cervenka Consulting Prague Czech Republic 23
17. ee da Peake hae e Pde al at 12 5 2 FReET A version o ed ataca o Oe a be 13 6 LATIN HYPERCUBE SAMPLING 13 OL General datay o racea p 4a RA A E e A ee 13 6 2 Check samples 5 2 eyes eee Gow we ep ee nae ee boby s oa a a 14 6 3 Model nal sis san ke ee we ae ep ee ae ee ks 16 7 SIMULATION RESULTS ASSESSMENT 17 fel gt Histograms lt saree i A ee gos A A AA 17 7 2 Limit state function LSF definition o a Sik EE edk 17 fod Sensitivity ANALYSIS 1x eb de a EG by ee eae k Ale one kc 18 7 4 Reliability analysis lt lt lt 444 ee 21 1 INTRODUCTION 1 INTRODUCTION The purpose of this manual is to provide a full description of the graphic user interface of program FReET This document is compatible with FReET version 1 4 released in July 2006 The multipurpose probabilistic software FReET has been developed for sta tistical sensitivity and reliability analysis of both simple and computationally intensive user defined engineering problems An emphasize is on small sample reliability techniques which use very small number of simulations The main aim of the software is to enable a probabilistic treatment of complex engineering problems coded into deterministic soft ware where classical reliability approaches are not feasible The software is designed in the form suitable for relatively easy probabilistic assessment of any user defined problem The name of the software reflects this strategy FReET is the acronym for Feasi
18. ensitivity analysis LA Reliability Figure 3 Main program tree FREET PROGRAM DOCUMENTATION USER MANUAL 4 STOCHASTIC MODEL 4 1 Random variables The window Random Variables Fig 4 allows the user friendly input of basic random variables of analyzed user defined problem Uncertainties are modelled as random vari ables described by their probability density functions PDF Every random variable has the name and is described by theoretical probability distribution and statistical character istics statistical parameters or by combination of characteristics and parameters button Descriptors The user can select from the set of selected theoretical models like normal lognormal Weibull rectangular etc The model is selected from the list of distributions which will appear when clicking on Distribution Fig 6 They are ordered approximately according to the expected frequency of the use for practical problems Both 2 parametric and 3 parametric are included Note that also negative forms of some distributions are included Following probability distributions are included e Deterministic e Normal e Lognormal 2par e Lognormal 3par e Weibull min 2par e Weibull min 3par e Weibull max 2par e Weibull max 3par e Raileigh e Raileigh negative e Beta 4par e Gamma 2par e Gamma negative 2par e Gamma 3par e Gamma negative 3par e Exponential e Exponential negative e Gum
19. esults are collected M version Analyzed user defined function defined by equation interpreter or using the DLL function unit is repeatedly solved here First the user has to use double click on New Model Function definition a new line will appear Then there are two possibilities to define response limit state functions The button a b will activate the equation interpreter window Fig 12 This option can be used for simpler functions only For more complicated functions the concept of DLL function has to be used as described in section 5 1 By clicking on button FReET will require to input name of DLL function which contains response limit state function The names of internal functions programmed inside of DLL function are indicated on the screen Exported functions Fig 19 Run Model Analysis will finally start real simulation Note that not only one function can be defined here FReET allows to define several functions here and in consequent step simulation to treat them simultaneously Every function will use the same set of randomly generated realizations of random variables they are common for all functions The user should take into account overall time of whole simulation in case of computationally demanding response limit state functions A version All random solutions of nonlinear fracture mechanics analysis are per formed via SARA Studio environment and output files of ATENA numerical outputs are
20. iability index Cornell 3 and corresponding failure probability Cornell pp based on estimation of statistics of safety margin on the assumption of normal probability distribution for safety margin 4 Mean Std Failure probability estimation CF p based on the selection of the most suitable theoretical model for PDF of safety margin curve fitting approach Calculation of failure probability based on classical freguency definition of probabil ity N Ntot where Ny is number of realizations resulting in a failure negative limit state function and N o is total number of simulations Note that this alternative can be used only for very large number of simulations Accuracy of this estimation is expressed by coefficient of variation of this frequency estimate COV pr Software FReET is generally designed for small numbers of simulations this classical Monte Carlo based estimation of failure probability is included here mainly for reference studies An utilization is related to the usage of crude Monte Carlo simulation using large numbers of simulations A dol Elle Edt Yiew Help D a A 2 7 ELH Stochastic model limit 1 la Random variables Soff Statistical correlation Latin Hypercube Sampling 0 004 Std sults Assessmer Mean we Fr 0 0035 0 003 0 0025 0 002 0 0015 0 001 0 0005 100 D 100 Digite Drawing 3 Distribution details 4 POF C CDF Result name Classes M
21. ing applied consequently does not require this strong requirement the feature is theoretically described in FReET Part 2 Theory Program FReET will prepare the sampling plan with correlation matrix as close as possible to the target correlation matrix but positive definite As the knowledge of correlation is always pure and the user can have problems to input a correlation matrix which is positive definite this feature can be considered to be an advantage from practical point of view 10 FREET PROGRAM DOCUMENTATION USER MANUAL Eideska fre Freet E lnx Fie Edit Mew Help DSM AA gt SHH Stochastic model Correlation matrix image la Random variables positive definite HE Statistical conelation E Latin Hypercube Sampling fi General Data 1 ele Check samples BB Model Analysis Elen Simulation Results Assessme m s lin Histograms ha LSF definition 200 YE Sensitivity analysis la Reliability azs Correlation coeficient Peason Epeaman Load Resistance Comparative values All variables Load a Load F Resistance b Resistance Rsd Resistance Rbd Load a 4 05 0 a 0 Load F 0 5 a Resistance b o a o 1 a o 0 1 o 0 Resistance Rsd o Figure 11 Window Statistical Correlation 5 RESPONSE LIMIT STATE FUNCTION DEFINITION In spite of the fact that response limit state function selection or and definit
22. ion is realized in the part Latin Hypercube Sampling Model analysis the description of possibilities is described here The reason is that it is a fundamental step more or less related to stochastic model definition 5 1 FReET M version 5 1 1 Equation interpreter For analysis of a simple response limit state functions an equation interpreter has been created The activation of the interpreter window Fig 12 is done in Model analysis using a b button The list of all basic random variables which were included in Sto chastic model will be listed on the right The function itself is created using variables ID 1 2 etc selected by double click and pads Functions and Numeric For func tion debugging and checking purposes button Test will provide the result of function evaluation with mean values 11 5 1 FReET M version 5 RESPONSE LIMIT STATE FUNCTION DEFINITION Edit response limit state function x alx 235 x4x6 gafel D Clear Delete Test Functions M Numeric Variables sin an sinh anh l x1 Load q ud Load F cos acos cash acosn 8 g i 3 Eacictanes ast x4 Resistance b tan atan tn atan 4 5 6 x5 Resistance Asd 6 Resistance Abd log exp coro 10 1 2 3 i sqrt Ka tad deg 0 s o abs oc ceil hypo e pi 1 Cancel OK Figure 12 Win
23. ion matrix as close as possible Fig 15 Basic parameter number of simulations is on input here Random input parameters are generated according to their PDF using Monte Carlo type simulation and generated realizations of random parameters are then used as inputs for analyzed function com putational model The solution is performed repeatedly in following Model analysis and results structural responses are saved Four alternatives of sampling scheme can be selected sampling type LHS probabilistic means preferable alternative LHS probabilistic median LHS random and pure Monte Carlo details are provided in Theory guide of FReET 13 6 2 Check samples 6 LATIN HYPERCUBE SAMPLING Simulated annealing is used as most powerful technique to impose statistical corre lation A heuristic time prediction is included the estimation is rough in special cases the real time could be very different Parameters of simulated annealing are estimated as suitable defaults a recommended robust setting but the user can change them The process of the samples generation is initiated by button Run Imposing of statistical correlations could be stopped by the button Stop Sampling The window with informa tion about achieved accuracy deviations of elements in correlation matrix is controlled desired and obtained is displayed after simulated annealing process Fig 16 The result of this step is the table of random realization
24. it FReET program can be terminated by choosing this menu item If data contents in FReET were changed and not saved before a dialog box will appear and makes possible a data saving before exiting the program 3 MAIN PROGRAM TREE Main program tree is located in the left field of the program window It represents main features key entries of the program guide the user when using the program 3 MAIN PROGRAM TREE Ulo it T a Benchmark e EB M zew souboru Benchmarkt tre Ulo it jako typ Freet Files fre Stormo A Figure 2 Save FReET file e Stochastic Model Basic random variables of the problem are defined here by statistical moments or and statistical parameters including theirs statistical correlation e Latin Hypercube Sampling Sampling is performed here First statistical correlation is imposed by simulated annealing approach Sampled realizations can be visually and numerically checked Second model analysis is performed using prepared samples of random variables e Simulation Results Assessment Statistical sensitivity and reliability analysis is performed based on sampling Ad ditional limit state function definitions can be defined also here Stochastic model La Random variables mi Statistical correlation E Latin Hypercube Sampling General Data 5d Check samples mf Model Analysis lm Simulation Results Assessment Jin Histograms R E LSF definition S
25. ity sensi Category 1 45 0 741 0 741 Category 1 X6 0 508 Limit State Function 1 0 12 Category 1 X3 0 197 Category 1 X2 0 171 Category 1 1 0 122 Category 1 X4 Figure 22 Window Sensitivity Analysis positive sensitivity parallel coordinates 0 073 19 7 3 Sensitivity analysis 7 SIMULATION RESULTS ASSESSMENT S Benchmark4 fre Freet Bile Edit view Help Daa we S E TM Stochastic model La Random variables FE Statistical correlation E Latin Hypercube Sampling ES General Data ool Check samples AB Model Analysis Eres Simulation Results Assessment Wla Histograms R E LSF definition Sensitivity analysis LA Reliabilty Paralel Coordinates Cartesian Coordinates I Numbers Mame sensi a 0 508 sensis sensiy Category 145 Category 1 X6 Category 1X3 Category 1 X2 Category 1 X1 Category 1 44 Limit State Function 1 Figure 23 Window Sensitivity Analysis negative sensitivity parallel coordinates AZ z o Ele Edit view Help Don mese ET Stochastic model La Random variables Ef Statistical correlation EX Latin Hypercube Sampling HP General Data Check samples HB Model Analysis Elas Simulation Results Assessment olay Histograms R E LSF definition mz Sensitivity analysis
26. ls is not limited The text file of the database named 4 2 Statistical correlation 4 STOCHASTIC MODEL Level 1 F Material FLeval 2 Concrete 1 1 1 Concrete JC55 Level 5 Ready mixed Level 6 cls tlevel 7 1111 compressive strength MPa inean Level 7 1111 Tensile strength mPa mean 2 95 std 0 93 bistri Level 7 1111 Modulus of elasticity mPa L Level 7 1111 Figure 10 Database internal structure in ASCII text file DATABASE txt should be in the same directory as FReET program It can be updated modified or replaced according to the need of particular application The flexible structure of database file is shown in Fig 10 4 2 Statistical correlation The window Statistical Correlation serves for the input of statistical correlation among random variables described by correlation matrix Fig 11 The user can work at the level of subset of correlation matrix one group of random variables or at the global level all random variables forming a large global correlation matrix All variables Statistical correlation among the variables is imposed using simulated annealing in subsequent step The type of correlation coefficient can be selected Pearson or Spearman The level of correlation during interactive input is highlighted in an upper window Fig 11 the basic feature of correlation matrix is checked if matrix is positive definite or not Note that the simulated anneal
27. lysis Ella Simulation Results Assessment lay Histograms R E LSF definition m Sensitivity analysis Lo Reliability New LSF Delete LSF LSF Mame Classes Operation ji Limit State Function 1 15 Ready m Figure 21 Response R action of loading E safety margin Z R E 7 3 Sensitivity analysis The window Sensitivity analysis shows the importance of random variables Nonpara metric rank order correlation coefficients are calculated between all random input variables and response variables A high positive correlation coefficient e g 0 8 indicates that the response or limit state function is very sensitive to that particular variable bigger vari 18 FREET PROGRAM DOCUMENTATION USER MANUAL able bigger response A high negative correlation coefficient e g 0 8 indicates opposite strong sensitivity bigger variable smaller response Positive and negative sensitivity is shown in separate columns There are two ways of graphical representation cartesian and parallel coordinates representations Parallel coordinate representation provides an insight into analyzed problem Random variables are ordered with respect of the sensitiv ity expressed by nonparametric rank order correlation coefficient Positive and negative sensitivity is shown in Fig 22 and Fig 23 parallel coordinates and in Fig 24 and Fig 25 cartesian coordinates Note 1 When the user assess this relative measure of sensi
28. mechanics for concrete is integrated fully using specially developed software environment called SARA Studio It enables communication between FReET and ATENA software SARA Studio is described fully in different documentation here only basic concept is outlined Response variable is selected at the level of ATENA deterministic model It is asso ciated with definition of monitoring points and assigned quantities Response function represents ATENA computational modeling response is a quantity at monitoring point Typically it is a peak load of load deflection diagram or maximum deflection or maximum crack width Variables at monitoring points represent responses and they are transferred into FReET software Response variables from ATENA can be evaluated statistically in part Simulation results assessment including sensitivity analysis For reliability analysis these response variables have to be combined with additionally defined comparative values defined as additional variables in Stochastic model Number of values associated with monitoring points can be combined with number of comparative values It enables definition of limit state functions representing ultimate and serviceability limit states This combination is described in details in section 7 6 LATIN HYPERCUBE SAMPLING 6 1 General data Latin hypercube samples are prepared first samples are reordered by simulated anneal ing approach in order to match required correlat
29. s of elasticity k Ultimate strain X F Apply h draft Part 3 Material JESS Joint Committee on Structural Safety 12 Properties Kersken Bradley M Rackwitz A Stochastic Modeling of Materials Properties and Quality Control JESS Working Document IABSE publication March 1991 Rackwitz FR Predictive Distribution of Figure 9 Database support of statistical parameters for random variables of SARA Studio Additional random variables can be defined by New variable button to form limit state function in the category Comparative values This is necessary if reliability analysis is planned ATENA provides response function maximum capacity corresponding to peak load deflection or crack width and these quantities should be compared with load maximum allowable deflection or maximum crack width in order to form limit state function In order to support the input of statistical characteristics for basic random variables the possibility to work with user defined database was worked out The user can click on Database button of Distribution support calculation and the hierarchical structure according to the database file will appear Fig 9 The user can go throw the structure to search for characteristics If they are found the user can transfer them automatically into random variable description table The structure of database file is self explanatory the number of hierarchical leve
30. sessment we can still decide to go back to Comparative values definition in Stochastic model In such case of new comparative definition it is not necessary to perform a time consuming simulation again 4 1 Random variables 4 STOCHASTIC MODEL Button Comparative values only should by selected in General Data window Then comparative values will be only sampled and the user can skip Model analysis and to go quickly to Simulation Results Assessment Distribution Deterministic E Lognormal Weibull min EV TIT Weibull max Ev III Rayleigh Figure 6 Combo box for selection of probability distribution x Moments Mean oso Sd fis Skew jos Kurt osa r Distribution 30 5 29 6 35 9 34 6 25 3 27 9 a 25 9 32 4 33 3 30 0 Normal SL 0 712 From file Calculate parameters Apply Figure 7 User defined distribution There is additional option for calculation at the level of selected PDF model Distrib ution support calculation and Details buttons allow recalculation of characteristics and parameters probabilities percentiles etc Fig 8 This probability distribution calculator enables to have overall numerical information on selected distribution M version Basic random variables related to response limit state function should be defined in increasing order Category 1 Variable 1 2 etc Category 2 Variable 1 2 etc
31. tivity it is necessary to take into account the signs of input random variables considered in Stochastic model Therefore an option to change the sign of variable 171 is included here This option is recommended to use in case the negative signs are considered for some basic input random variables in Stochastic model for correct understanding of sensitivity results Note 2 What if study known as deterministic sensitivity analysis can be done easily by FReET To study absolute influence of a specific input variable we can consider it as random variable with rectangular distribution and to keep other variables as determin istic Only this one variable is then sampled consequent model analysis will provide a clear influence of this variable Sensitivity window in cartesian coordinates will provide just functional relationship between variable varying between upper and lower limits of rectangular distribution and response variable Fig 26indicates a linear influence of a variable In x File Edit Mew Help Dem eee atin amping F General Data M heck samples E Model Analysis Asse mz Sensitivity analysis Los Reliability A is y WU i h AN i if A AW YAR n a l my j w i M i AN i i A n NY V Al K n j i i AN H NAL N Mi w N A lumbers E Parallel Coordinates Cartesian Coordinates Name seri sensi sens
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