Basis variable

Basics

A basic random variable is a stochastic model of a quantity/value used in limit state functions. The stochastic model is described by a distribution function and the respective parameters. Such parameters influence the characteristics of the function. Usually one parameter shifts the position of the high point and another parameter shifts the width of the high point. It shall be noted that multiparameter distribution functions are also available. The position and the width can be described by the respective moment statistics (i.e., expected value and standard deviation). Normally, transformation equations are available to derive appropriate moment statistics and remaining input parameters for the following distribution types:

Normal distribution (norm) Lognormal distribution (lnorm) Gumbel distribution (gumbel) (Package: EVD) Gamma distribution (gamma) *Weibull distribution (weibull)

For the characterisation of each distribution function, it is sufficient to specify either the expected value and the standard deviation or the function parameters, since the missing pair of values can be automatically determined. For example, when specifying the expected value and the standard deviation, the covariance is automatically calculated or, on the other way around, a standard deviation is calculated on the basis of the mean value and the covariance. For unknown distribution functions, it is necessary to specify the parameters and, if simulated with Monte-Carlo Simulation with Importance Sampling (MC-IS), also the mean value and standard deviation are needed.

PROB_BASEVAR(Id, Name, Description, Package, DistributionType, DistributionParameters, Mean, Sd, Cov, X0)

var.f_ck <- PROB_BASEVAR(Id=1,Name="f_ck",Description="Char. Betondruckfestigkeit",Package="TesiproV",                         DistributionType="lt",DistributionParameters=c(3.85,0.09,3,10),Mean=47.35,Sd=5.86)

PARAM_BASEVAR(Id, Name, Description, Package, DistributionType, Sd, ParamType, ParamValues)

var.d <- PARAM_BASEVAR(Id=1,Name="d",Description="Stat. Nutzhöhe",DistributionType="norm",Sd=10,ParamType="Mean",ParamValues= c(seq(200,950,50),seq(1000,3000,100))+10)

var.gamma_c <- PROB_DETVAR(Id=1,Name="gamma_c",Description="TSB Beton", Value=1.5)

var.V_Ed <- PARAM_DETVAR(Id=1,Name="V_Ed",Description="Einwirkende Querkraft", ParamValues=c(1,2,3,4,seq(5,10,0.5)))

Limit state function (LSF)

A limit state function can be formulated either as an objective function (function()) or as an arithmetic expression (expression()). Internally, the arithmetic expression is then converted into an objective function. Therefore, this input option enables better usability. In objective functions, all possible programmatic operations can be carried out (i.e., iterations or loops, conditions, optimizations). Accordingly, for simulation methods, they must be neither necessarily continuous nor differentiable. However, if the Mean Value First Order Second Moment (MVFOSM) method is used, the function must be differentiable at the point of the mean values of the input values (i.e., finite difference method).

Variables are used as input values for the functions. The naming in the function header (function(E, R)) must fit to the use in the function belly (function(E,R){R-E}). Also, the naming must match the name of the variable itself. Therefore, a list() with all used PROB_BASEVAR() objects is passed to the SYS_LSF() object (note: the objects with their variable names, here var1 and var2, are to be passed, not the variable name itself). In the case of multiple limit state functions being used, the function can be given a name so that it can be better later assigned in printouts. An objective function() can then be defined via the $func() property. Alternatively, $expr() can be used to assign an expression(R-E). By means of $check() an internal check run is performed, which ensures that all the base variables are complete and suitable.

var1 <- PROB_BASEVAR(Name="E",Description="Effect",DistributionType="norm",Mean=10,Sd=1)
var2 <- PROB_BASEVAR(Name="R",Description="Resistance",DistributionType="norm",Mean=15,Sd=1.5)


lsf<-SYS_LSF(vars=list(var1,var2),name="LSF XY")
lsf$func <- function(E,R){R-E}
# you can run this check to see if all transformations of the variables worked, all needed data is available and the set of vars fits to the limit state function
lsf$check()

Algorithms

In this package, three different levels (or types) of reliability assessment methods are defined in line with the current European standardization:

• Level 1 (partial safety factors)
• Level 2 (MVFOSM, FORM, SORM)
• Level 3 (MC_CRUDE, MCIS, MCSUS)

Levels 2 and 3 can be implemented by means of this package (in order to validate Level 1 procedures, if necessary). The accuracy of the calculated failure probabilities - and ultimately, reliability indexes - increases through the levels.

A solution-oriented approach is always modeled in a so-called PROB_MACHINE(), i.e. an object that stands for a solution algorithm. Over the property fCall, the following reliability assessment methods are available:

Level 2 Methods: * Mean Value First-Order Second Moment (MVFOSM) * Advanced First-Order Reliability Method (FORM) * Second-Order Reliability Method (SORM)

Level 3 Methods: * Crude Monte-Carlo Simulation (MC_CRUDE) * Monte-Carlo Simulation with Importance Sampling (MC_IS) * Monte-Carlo Simulation with Subset Sampling (MC_SUS)

The property $name serves again for better identification in the later output protocol. A list() object in the $options property can then be used to specify special options according to the methods.

machine <- PROB_MACHINE(name="MVFOSM",fCall="MVFOSM",options=list("isExpression"= FALSE,"h"=0.1))

machine <- PROB_MACHINE(name="FORM Rack.-Fieß.",fCall="FORM",options=list("n_optim"=20, "loctol"=0.001, "optim_type"="rackfies"))

machine <- PROB_MACHINE(name="FORM Lagrange.",fCall="FORM",options=list("optim_type"="auglag"))

machine <- PROB_MACHINE(name="SORM",fCall="SORM")

machine <- PROB_MACHINE(name="MC CoV 0.05",fCall="MC_CRUDE",options=list("n_max"=1e6, "cov_user"=0.05))

machine <- PROB_MACHINE(name="MC IS",fCall="MC_IS",options=list("cov_user" = 0.05, "n_max"=300000))

machine <- PROB_MACHINE(name="MC Subset Sampling",fCall="MC_SubSam",options=list("variance"="uniform"))

System object

The system object combines all the previous objects and organizes the correct calculation of the mathematical model. To this, there are two objects available:

• SYS_PROB Classical calculation of n systems with m algorithms.
• SYS_PARAM Parameterized calculation of one system with m algorithm

The SYS_LSF object already contains a list of the required basic variables. However, the system object still requires a list of functions to be calculated and the list of algorithms to be used. In addition, the $sys_type property can be used to specify whether the equations have a series $sys_type="serial" or parallel $sys_type="parallel" relationship.

The calculation process is then triggered with $run_machines(). In the console, the user is informed about the current state of the calculation. After completing the calculation, the results can be retrieved via the SYS_PROB object (in the example below ps). For more information see result interpretation.

You can set a debug.level field of SYS_PROB to 2, to get more information during the calculation, by default that value is zero.

var1 <- PROB_BASEVAR(Name="E",Description="Effect",DistributionType="norm",Mean=10,Sd=1)
var2 <- PROB_BASEVAR(Name="R",Description="Resistance",DistributionType="norm",Mean=15,Sd=1.5)


lsf<-SYS_LSF(vars=list(var1,var2),name="LSF XY")
lsf$func <- function(E,R){R-E}
# you can run this check to see if all transformations of the variables worked, all needed data is available and the set of vars fits to the limit state function
lsf$check()


machine <- PROB_MACHINE(name="FORM Rack.-Fieß.",fCall="FORM",options=list("n_optim"=20, "loctol"=0.001, "optim_type"="rackfies"))

ps <- SYS_PROB(
  sys_input=list(lsf),
  probMachines = list(machine),
  debug.level=0
)

ps$runMachines()


# Systemberechnungn (System aus zwei gleichen LSF´s)
ps2 <- SYS_PROB(
  sys_input=list(lsf,lsf),
  probMachines = list(machine),
  sys_type="serial"
)

ps2$runMachines()
ps2$calculateSystemProbability()

#ps2$calculateSystemProbability("MCSUS")

#params <- list("cov_user"=0.10)
#ps2$calculateSystemProbability("MCC",params)
#ps2$calculateSystemProbability("MCIS",params)

var1 <- PARAM_BASEVAR(Name="E",Description="Effect",DistributionType="Norm",Sd=1,ParamType="Mean",Values=c(9,11,0.1))

#[...]

ps <- SYS_PARAM(
  sys_input=list(lsf),
  probMachines = list(form)
)

ps$runMachines()

Interpretation of results and post processing

After the calculation is completed, the results are available in the PROB_SYS object. For regular calculations (PROB_SYS) the following results can be retrieved:

• Simple calculation
  • beta_single
  • res_single
• System calculations
  • beta_sys
  • res_sys

Here beta_* is the beta values summarized while res_* is the full result output via list(). The list $res_* is structured as follows ps$res_*[[lsf]][[machine]]$value.

For parametric calculation stands: * beta_params * res_params

where the first list level of $res_params corresponds to the parameter iterations. The data can be then processed individually.

In addition, the results can be printed to a file using the command $printResults(filename), where filename corresponds to the file name and the file is stored in the current working directory (see getwd() or setwd()).

The project can also be stored differently to allow later postprocessing in other files. For this, the command $saveProject(level,filename) is used. The following stages describe a different level of detail:

• 1. Only the beta values are stored.
• 2. The $res_* object is saved.
• 3. The entire ps object is stored.
• 4. An image of the current working memory is created, which can be reloaded later.

Stages 1 to 3 can be then read back into working memory using the readRDS(filename) command, while Stage 4 is reloaded using load(filename).

ps <- SYS_PARAM(
  sys_input=list(lsf),
  probMachines = list(form)
)

ps$runMachines()

local_beta <- ps$beta
local_runtime <- ps$res_single[[1]][[1]]$runtime

#define path with setwd("...")
ps$printResults("result_file") #Prints a report file in .txt format with all informations about the calculation
ps$saveProject(4,"project_file") #stores the project in a file

Two normal distributed variables

A limit state equation 20-(x1-x2)^2-8*(x1+x2-4)^3 is to be investigated, where x1 and x2 are normally distributed around the expected value of 0.25 and the standard deviation of 1. The example is taken from the documentation of the UQLab software.

Firstly, two basic variables are created and assigned to the R variables Var1 and Var2. The first variable is named as “X1” and the second as “X2”. These names are then reused in the definition of the limit state function (i.e., to compare the input variables of the function). In addition, both variables receive the indicator “norm” as distribution type (DistributionType) and the mean value defined above or the corresponding standard deviation.

Subsequently, a limit state function object (SYS_LSF) is assigned to the R variable lsf. A list of variables (Var1,Var2) is passed to the object and a name is assigned for later identification. In addition, the corresponding limit state equation is defined via the object property “func”.

The R variable “FORM” is assigned a solution method of the Level 2 (FORM) without any additional setting parameters.

Finally, the modules are merged in the SYS_PROB object and the R variable “ps” and the calculation is started by means of the $runMachines() method. In the following result log the intermediate results of the individual iteration steps are shown. Using the “ps” object, different results (here, for example, the beta value) can be retrieved after a successful calculation.

Var1 <- PROB_BASEVAR(Id=1,Name="X1",DistributionType="norm",Mean=0.25,Sd=1) #kN/m²
Var2 <- PROB_BASEVAR(Id=2,Name="X2",DistributionType="norm",Mean=0.25,Sd=1) #m

lsf<-SYS_LSF(vars=list(Var1,Var2),name="UQLab 2d_hat")
lsf$func <- function(X1,X2){
  return(20-(X1-X2)^2-8*(X1+X2-4)^3)
}

form<-PROB_MACHINE(name="FORM Rack.-Fieß.",fCall="FORM")

ps <- SYS_PROB(
  sys_input=list(lsf),
  probMachines = list(form)
)
ps$runMachines()

ps$beta_single
#>                  UQLab 2d_hat
#> FORM Rack.-Fieß.     3.434565

Non-normally distributed basic variable

To demonstrate non-normally distributed basic variables, an example from Nowak & Collins (Example 5.11, Page 127/128) is calculated. Here normally distributed, lognormally distributed as well as gumbel distributed variables are used. While “norm” and “lnorm” are still contained in the standard scope of R (package “stats”) the Gumbel distribution is to be considered on the basis of the EVD-package (EVD := extreme value distributions).

library("evd")
X1 <- PROB_BASEVAR(Id=1,Name="Z",DistributionType="norm",Mean=100,Cov=0.04) #kN/m²
X2 <- PROB_BASEVAR(Id=2,Name="Fy",DistributionType="lnorm",Mean=40,Cov=0.1) #m
X3 <- PROB_BASEVAR(Id=2,Name="M",DistributionType="gumbel",Package="evd",Mean=2000,Cov=0.1) #m

lsf<-SYS_LSF(vars=list(X1,X2,X3),name="Nowak & Collins Exp5.11_1")
lsf$func <- function(M,Fy,Z){
  K <- Z*Fy
  return(0.9*K-M)
}

form<-PROB_MACHINE(name="FORM Rack.-Fieß.",fCall="FORM")

ps <- SYS_PROB(
  sys_input=list(lsf),
  probMachines = list(form)
)
ps$runMachines()

ps$beta_single
#>                  Nowak & Collins Exp5.11_1
#> FORM Rack.-Fieß.                  5.388625

Multiple limit state functions in one system

Basically, the “SYS_PROB” object is able to calculate several LSF’s by means of Level 2 as well as Level 3 methods. This can be used if a system consists of several limit state functions and if the system safety is to be determined by means of so-called SimpleBounds. In this case, the failure probability must be determined for each individual limit state equation and then calculated to a system failure probability. On the other hand, the development of a limit state function often poses several variants in equations or variables, so that by the simultaneous calculation of several equations satisfactory comparability is achieved and a high usability remains granted.

If, however, the system failure probability is not to be estimated by means of the SimpleBounds, but is to be approximated by simulation methods (e.g., MC-Crude), a system must also be constructed according to the following scheme.

In this example, the above examples are first determined individually as a system using a FORM calculation and then the system failure probability is determined using different methods.

Var1 <- PROB_BASEVAR(Id=1,Name="X1",DistributionType="norm",Mean=0.25,Sd=1) #kN/m²
Var2 <- PROB_BASEVAR(Id=2,Name="X2",DistributionType="norm",Mean=0.25,Sd=1) #m

lsf1<-SYS_LSF(vars=list(Var1,Var2),name="UQLab 2d_hat")
lsf1$func <- function(X1,X2){
  return(20-(X1-X2)^2-8*(X1+X2-4)^3)
}


X1 <- PROB_BASEVAR(Id=1,Name="Z",DistributionType="norm",Mean=100,Cov=0.04) #kN/m²
X2 <- PROB_BASEVAR(Id=2,Name="Fy",DistributionType="lnorm",Mean=40,Cov=0.1) #m
X3 <- PROB_BASEVAR(Id=2,Name="M",DistributionType="gumbel",Package="evd",Mean=2000,Cov=0.1) #m

lsf2<-SYS_LSF(vars=list(X1,X2,X3),name="Nowak & Collins Exp5.11_1")
lsf2$func <- function(M,Fy,Z){
  K <- Z*Fy
  return(0.75*K-M)
}

form<-PROB_MACHINE(name="FORM Rack.-Fieß.",fCall="FORM")

ps <- SYS_PROB(
  sys_input =list(lsf1,lsf2),
  sys_type = "serial",
  probMachines = list(form)
)
ps$runMachines()

ps$beta_single
#>                  UQLab 2d_hat Nowak & Collins Exp5.11_1
#> FORM Rack.-Fieß.     3.434565                  3.710911

ps$calculateSystemProbability()
ps$beta_sys
#>            [,1]
#> SB min 3.352809
#> SB max 3.434565

The following lines are not executed because the calculation time is several minutes depending on the machine. Results in the range of the above interval are to be expected. The results can be retrieved after the calculation using “$beta_sys”.

params <- list("cov_user"=0.05,"n_max"=50000)
ps$calculateSystemProbability("MC",params)

params <- list("cov_user"=0.05,"n_max"=50000)
ps$calculateSystemProbability("MCIS",params)

ps$calculateSystemProbability("MCSUS",params)

Parameterstudy

• Punching according to DIN EN 1992-1 + NAD
• Effective depth (var.d) is parameterized (cf. vector at “ParamValues”)
• Additional parametric deterministic variable “var.V_Ed” which depends on "pre.d
• Calculation by FORM and postprocessing see below.

var.d <- PARAM_BASEVAR(Id=1,Name="d",Description="Stat. Nutzhöhe",DistributionType="norm",Sd=10,ParamType="Mean",ParamValues= c(seq(200,800,50),seq(1000,3000,100))+10)
pre.d <- c(seq(200,800,50),seq(1000,3000,100))

var.f_ck <- PROB_BASEVAR(Id=1,Name="f_ck",Description="Char.Betondruckfestigkeit",Package="TesiproV",DistributionType="lt",DistributionParameters=c(3.85,0.09,3,10),Mean=47.35,Sd=5.86)
pre.f_ck <- 35

var.f_ywk <- PROB_BASEVAR(Id=1,Name="f_ywk",Description="Zugfestigkeit Bügelbewehrung",DistributionType="norm",Mean=560,Cov=0.02)
pre.f_ywk <- 500

var.f_yk <- PROB_BASEVAR(Id=1,Name="f_yk",Description="Zugfestigkeit Längsbewehrung",DistributionType="norm",Mean=560,Sd=30)
pre.f_yk <- 500

var.rho_l <- PROB_BASEVAR(Id=1,Name="rho_l",Description="Biegebewehrungsgrad",DistributionType="norm",Mean=0.01,Cov=0.02)
pre.rho_l <- var.rho_l$Mean

var.c_D <- PROB_BASEVAR(Id=1,Name="c_D",Description="Stützendurchmesser",DistributionType="norm",Mean=500+0.003*500,Sd=4+0.006*500)
pre.c_D <- 500

var.theta_c <- PROB_BASEVAR(Id=1,Name="theta_c",Description="Modellunsicherheit c nach DIBT",DistributionType="lnorm", Mean=1.0644,Cov=0.1679)

var.gamma_c <- PROB_DETVAR(Id=1,Name="gamma_c",Description="TSB Beton", Value=1.5)
var.gamma_s <- PROB_DETVAR(Id=1,Name="gamma_s",Description="TSB Stahl", Value=1.15)
var.alpha_cc <- PROB_DETVAR(Id=1,Name="alpha_cc",Description="Langzeitfaktor AlphaCC", Value=0.85)

V_Ed <- c(0.6412447,0.8760515,1.1424185,1.4399554,1.7554079,2.0188587,2.3003489,2.5997038,2.9167825,3.2514671,3.6036567,3.9732630,
          4.3602076,6.0800504,7.0424326,8.0726216,9.1702994,10.3351829,11.5670173,12.8655715,14.2306347,15.6620132,17.1595283,18.7230145,
          20.3523175,22.0472935,23.8078076,25.6337332,27.5249511,29.4813487,31.5028193,33.5892621,35.7405811,37.9566849)
var.V_Ed <- PARAM_DETVAR(Id=1,Name="V_Ed",Description="Einwirkende Querkraft", ParamValues=unlist(V_Ed))

lsf1 <-SYS_LSF(vars=list(var.d,var.f_ck,var.f_yk,var.f_ywk,var.rho_l,var.c_D,
                         var.theta_c,var.V_Ed,var.gamma_c,var.gamma_s,var.alpha_cc),
               name="Durchstanzwiderstand ohne Bewehrung")
lsf1$func <- function(d, f_ck,f_yk,f_ywk, rho_l, c_D, theta_c, V_Ed, gamma_c, gamma_s, alpha_cc){

  f_cd <- f_ck*alpha_cc/gamma_c
  f_yd <- f_yk/gamma_s
  f_ywd <- f_ywk/gamma_s

  u_0 <- c_D*pi
  u_1 <- (c_D/2+2*d)*2*pi
  u_0.5 <- (c_D/2+0.5*d)*2*pi

  k <- min(1+sqrt(200/d),2)

  if((u_0/d)<4){
    C_Rd_c <- 0.18 * (0.1*u_0/d + 0.6)
  }else{
    C_Rd_c <- 0.18
  }
  rho_l <- min(rho_l,0.5*f_cd/f_yd,0.02)
  C_Rd_c1 <- C_Rd_c * k * (100*rho_l*f_ck)^(1/3)

  v_Rd_c <- C_Rd_c1
  V_Rd_c <- v_Rd_c*d*u_1*1e-6*theta_c
  return(V_Rd_c-V_Ed)
}

form <- PROB_MACHINE(name="FORM RF.",fCall="FORM",options=list("n_optim"=20, "loctol"=0.0001, "optim_type"="rackfies"))

ps <- SYS_PARAM(
  sys_input=list(lsf1),
  probMachines = list(form)
)
ps$runMachines()

d <- pre.d
betas <- unlist(unname(ps$beta_params))
plot(x=d,y=betas,type="l")