Docstoc

e-stud.vgtu.ltusersfilesdest4722vaidogas_glam

Document Sample
e-stud.vgtu.ltusersfilesdest4722vaidogas_glam Powered By Docstoc
					ESTIMATION OF STRUCTURAL RELIABILITY
               BY MEANS OF
         MONTE CARLO METHOD
    Definition & calculation of reliability

                ERASMUS LECTURE, BEng LEVEL



                  Egidijus R. Vaidogas
    Vilnius Gediminas Technical University, Lithuania
                          Contents of the lecture
                               Three part talk

          PART I                      PART II                            PART III



                                                                  Estimation of
                                  Definition of
     Problem of                                                 failure probability
                              reliability & failure
 structural reliability                                            by means of
                                   probability
                                                               Monte Carlo method


       Max. 20 min                 Max. 40 min                        Max. 60 min




E.R.Vaidogas,VGTU, 2009        Erasmus Lecture at the University of Glamorgan       2
                       PART I
          The problem of structural reliability

E.R.Vaidogas,VGTU, 2009   Erasmus Lecture at the University of Glamorgan   3
                 What is the principal objective?
                  If you think that safety is expensive,
                         try to have an accident!




                                                                   Failures
                               Running
      Monetary
                               technical
       profit                   system




E.R.Vaidogas,VGTU, 2009      Erasmus Lecture at the University of Glamorgan   4
                          Phenomenon in focus 1/3
                      Unfortunately, structures fail …




           Partial collapse of                          Partial collapse of
           Pentagon Building                              CGA Terminal



E.R.Vaidogas,VGTU, 2009        Erasmus Lecture at the University of Glamorgan   5
                          Phenomenon in focus 2/3
                                Minor failures




     Failure due to debris impact



      Failure due to insufficient shear
                               capacity

E.R.Vaidogas,VGTU, 2009        Erasmus Lecture at the University of Glamorgan   6
                          Phenomenon in focus 3/3
                              Catastrophic failures




         Collapse of I-35W Mississippi River bridge, August 1, 2007
                           13 killed, 145 injured
E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan   7
                            Measuring reliability
                          Failures of FORMULA 1 cars
  TOP SPEED                                                                      CHANCE OF

  MASS                                                        FAIL-SAFE BEHAVIOUR
                                                                           (RELIABILITY)
  AERODYNAMIC
  PROPERTIES




E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan          8
                          Problem of reliability 1/7
              Reliability theory: arguments in favor

 DETERMINISTIC analysis of
 structures, say, Eurocodes                             Assessment of
                                                  DURABILITY OF STRUCTURES


                                                    Incorporation of possibility of
                      Time-independent                    HUMAN ERRORS
                       PROBABILISTIC
                      (reliability-based)
                          analysis of                     Consideration of
                     error-free structures              STRUCTURAL SYSTEM
                                                   rather than individual components

                                                          Consideration of
                                                      ABNORMAL SITUATIONS
                                                        (accidental actions)

E.R.Vaidogas,VGTU, 2009          Erasmus Lecture at the University of Glamorgan       9
                          Problem of reliability 2/7
               Reliability theory: arguments against



  The need to study probability calculus and statistics
  The need to collect statistical data on structures and actions
   (loads)
  The need to move outside the “safe and customary“ area
   ruled by design codes of practice
  Do you know the answer on the question “How safe is safe
   enough?” ?




E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan   10
                          Problem of reliability 3/7
    The risk of death as a result of structural failure*
                          Approximate death
                                              Estimated typical              Typical risk of
       Activity               rate ×10-9
                                             exposure (hr/year)             death ×10-6/ year
                          deaths/hr exposure
Construction work              70...200                   2200                   150...400
   Manufacturing                  20                      2000                        40
 Coal mining (UK)                 210                     1500                       300
   Building fires                1...3                    8000                       8...24
      Air travel                 1200                       20                        24
      Car travel                  700                      300                       200
     Train travel                 80                       200                        15
Structural failures              0,02                     6000                        0,1
* Melchers, R. E. (1987) Structural reliability Analysis and Prediction. Chichester: Ellis
Horwood/Wiley

E.R.Vaidogas,VGTU, 2009             Erasmus Lecture at the University of Glamorgan            11
                          Problem of reliability 4/7
 Why should we be concerned about structural reliability
  Individuals: involuntary of risk due to structural failures
The risk levels for buildings and bridges are usually associated with involuntary risk
and are much lower than the risk associated with voluntary activities (travel, mountain
climbing, deep see fishing)

  Society: failure results in decrease of confidence in stability and
   continuity of one's surroundings
 Society is interested in structural reliability only in the sense that a structural failure
 with significant consequences shatters confidence in the stability and continuity of
 one’s surroundings

  Engineers: the need to apply novel structures and novel construction
   methods generates interest in safety
 Design, construction, and use of new or particularly hazardous systems should be of
 particular interest in their safety (new and unique bridge, new off-shore structure, NPP,
 chemical plant, liquefied gas depot)
E.R.Vaidogas,VGTU, 2009            Erasmus Lecture at the University of Glamorgan          12
                          Problem of reliability 5/7
                 Principal causes of structural failures

                 Human errors & deliberate actions

 The main subject of
  reliability theory

                                  Factor of
                                 uncertainty

Accidental actions                                             Accumulation of
(explosions, collisions, etc.)                                damage (ageing)


E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan   13
                          Problem of reliability 6/7
       The need to bridge the gap: how to join quickly?
         STUDENTS & PRACTISING
                                                          ELITE SCIENTICS
              ENGINEERS




                                       GAP

E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan   14
                          Problem of reliability 7/7
                            The TERECO textbook
             PROBABILISTIC ASSESSMENT OF STRUCTURES USING
                            MONTE CARLO SIMULATION
                          Background, Exercises, Software
              (P. Marek, J. Brozzetti and M. Guštar, P.Tikalski, editors)
Publisher: ITAM CAS CZ Academy of Sciences of the Czech Republic, Prague,
                                    2001.
     The textbook with the CD-ROM is the final product of a pilot project
    sponsored by the Leonardo da Vinci agency, European commission,
                                  Brussels,
         "TERECO - Teaching reliability concepts using simulation",
                                 1999 - 2001.
                  For more information visit the website:
                         http://www.noise.cz/SBRA




E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan   15
                          THE END OF PART I




  Further talk:
   Mathematical definition of reliability & failure
    probability
   Estimation of the failure probability by means
    of Monte Carlo method




E.R.Vaidogas,VGTU, 2009      Erasmus Lecture at the University of Glamorgan   16
                          PART II
            Definition of Structural Reliability

E.R.Vaidogas,VGTU, 2009   Erasmus Lecture at the University of Glamorgan   17
                          The general principle 1/3
               Deterministic (non-probabilistic) analysis




   x1k               x1
                            Mathematical model                    ek                  e
                                 of structure
                                                                         rk ek
                x2k x2      (limit state function)

                            model uncertainty:
                               coefficient 
                                                                                 rk   r
         xnk         xn

E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan        18
                          The general principle 2/3
               Probabilistic (reliability-based) analysis
                                                                        Density of E


   x1k               x1                                                                e
                            Mathematical model                    e1k
                                 of structure
               x2k x2       (limit state function)                   Pf = P(R E)

                           model uncertainty:
                                                                          Density of R

                            random variable Xi

         xnk         xn                                                          rk   r

E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan         19
                          The general principle 2/3
                              Connecting point


     xi = material strength                                  xi = load



   0.05

   xid      xik                    xi                              xik           xid   xi


E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan          20
              Reliability & failure probability 1/10
                             Basic definition
                Reliability = 1 – failure probability
                                Ps = 1 – Pf

               Structure can either fail or survive:
                                Ps + Pf = 1

■ Failure * : an insufficient load-bearing capacity or inadequate serviceability
    of a structure or structural element.
■ Limit state: a state beyond which the structure no longer satisfies the
    design performance requirements.
* ISO 2394: 1998 (E). General principles on reliability for structures. ISO,
    Geneve, 1998.

E.R.Vaidogas,VGTU, 2009       Erasmus Lecture at the University of Glamorgan   21
              Reliability & failure probability 2/10
                  Reliability is usually not calculated!
     SURVIVAL OF STRUCTURE                        FAILURE OF STRUCTURE
       (fulfillment of specified                (exceedance of irreversible or
             requirements)                          reversible limit state)
            Ps = P(survival)                              Pf = P(failure)
                          P(survival) + P(failure) = 1
                                   Ps + Pf = 1

 Structure 1                 Ps1 = 0,9995                     Pf1 = 0,0005
 Structure 2                 Ps2 = 0,999                      Pf2 = 0,001

 Difference                  0,05%                            200%
E.R.Vaidogas,VGTU, 2009        Erasmus Lecture at the University of Glamorgan    22
              Reliability & failure probability 3/10
                          How safe is safe enough?

  Tolerable failure probabilities given in ENV 1991-1*
                               Tolerable failure                  Tolerable failure
      Limit state                 probability                       probability
                             (design working life)                   (one year)
     Ultimate                      723  10-7                           13  10-7
     Fatigue                 0,0668…723  10-7*                                 –

     Serviceability                  0,0668                             0,00135
* Depends on degree of inspectability, reparability, and damage tolerance.

 * ENV 1991-1: 1993. Basis of design and actions on structures. CEN, Brussels,
 1993.
E.R.Vaidogas,VGTU, 2009        Erasmus Lecture at the University of Glamorgan       23
              Reliability & failure probability 4/10
                    The way to the failure probability
          Statistical data on
   structural parameters and loads                     25,1; 33,2; 21,7; 30,3; ...


    Processing the statistical data
                                                                        x
    Mechanical model of structure

                                                                                x
   Estimation of failure probability                          no of failures
                          Pf                                   no of trials

E.R.Vaidogas,VGTU, 2009        Erasmus Lecture at the University of Glamorgan   24
              Reliability & failure probability 5/10
                           The role of data

      Statistical data reflects the ubiquitous uncertainty in
       structural parameters and loads.
      Statistical data is used to fit probability distributions of the
       structural parameters and loads.
      Statistical data determine the value of the failure probability
       Pf in the end.



    Data on material properties
    Data on geometrical quantities
    Data on direct actions (loads) and indirect actions
    Data on model uncertainties
E.R.Vaidogas,VGTU, 2009     Erasmus Lecture at the University of Glamorgan   25
            Reliability & failure probability 6/10
                          Processing the data
                                                   Estimation of distri 
       x1                                         
                                                   bution parameters:
       x2                                          x
       x3                           x             
                                                    2  s2
       x4           1 n                           
               x  n  xi                          Fitting a probability
                     i 1                         distribution
       xn  1           1 n                       
              s               xi  x 
                                             2 
       xn             n  1 i 1                  
                                                  
                     s
                 (100%)                         
               2009 x
E.R.Vaidogas,VGTU,
                                                    University of Glamorgan x
                                Erasmus Lecture at the                         26
              Reliability & failure probability 7/10
                           Vector of basic variables

         X = (X1, X2, ... , Xn); x = (x1, x2, ... , xn)
             random variables                    particular values

                          Visualisation of X and x

            The case n=3                                                 x3
           X = (X1, X2, X3)
    three-dimensional space                                               (x1,x2,x3)
             x = (x1, x2, x3)
          point in the space
                                                       x2                         x1
E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan         27
              Reliability & failure probability 8/10
                           Probability density function

 f(x1)                         f(x2)                                      f(xn)



                          x1                             x2
                                                                 ...                        xn

      univariate                   univariate                                  univariate
      PDF of      X1               PDF of X2                     ...           PDF of Xn

                       Joint PDF of basic variables

                       f(x) = f(x1) f(x2)  ...  f(xn)
  in the case that basic variables Xi are independent
E.R.Vaidogas,VGTU, 2009                Erasmus Lecture at the University of Glamorgan       28
               Reliability & failure probability 9/10
        It’s nothing more than uncertainty propagation!

          Pf = P( Random Safety Margin  0) =
                                        = P( g(X)  0 )
    UPPER-LEVEL
    UNCERTAINTY
                                               MECHANICAL                        LOWER-LEVEL
                                                 MODEL                           UNCERTAINTY
                              UNCERTAINTY PROPAGATION


•   Limit state function: a function g of the basic variables, which characterizes a limit state when
    g(x1, x2, ... , xn) = 0; g > 0 identifies with the desired state and g < 0 with the undesired state
    [state beyond the limit state].
•   Basic variable Xi : a part of a specified set of variables, X1, X2, ... , Xn, representing physical
    quantities which characterize actions and environmental influences, material properties
    including soil properties, and geometrical quantities.

E.R.Vaidogas,VGTU, 2009                Erasmus Lecture at the University of Glamorgan                     29
             Reliability & failure probability 10/10
      The problem is an integral, not the reliability itself!


        Pf = P( g(X)  0 )   ...  f X ( x ) d x
                                           Df

            Failure domain                                        Joint PDF

      Df  x | g ( x )  0                                      x1 , x2 , x3 , ...

          The role of model                                  The role of data




E.R.Vaidogas,VGTU, 2009      Erasmus Lecture at the University of Glamorgan            30
                          THE END OF PART II
                How to evaluate the multiple integral?

               Pf = P( g(X)  0 )   ...  f X ( x ) d x
                                                 Df


   Exact analytical methods
   Classical methods of numerical integration
   Approximate analytical methods (FORM/SORM methods)
   Monte Carlo method




E.R.Vaidogas,VGTU, 2009      Erasmus Lecture at the University of Glamorgan   31
                                  PART III
                          Monte Carlo Method

E.R.Vaidogas,VGTU, 2009       Erasmus Lecture at the University of Glamorgan   32
                          Monte Carlo method 1/4
       Failure probability as a mean of random variable

          1 if g ( x )  0 ( failure )
          
                                                      mass Ps               mass Pf
 I ( x)  
           0 if g ( x )  0 ( survival )
                                                           0                         1


                                                    1
Vector of random arguments X
                                                                  Ps
                                                                         Pf
Probability mass function of I(X)
                                                    0
                                                        0                         1       I(x)

 Pf  P ( I ( X )  1)   ...  I ( x ) f X ( x ) d x  E(I(X))
                             all x

E.R.Vaidogas,VGTU, 2009          Erasmus Lecture at the University of Glamorgan                  33
                            Monte Carlo method 2/4
                          Estimate of failure probability
   Intuitive                         Number of failures N f
  definition              P fe 
                                       Number of trials N
                                     1 N
                                        I(x j )
   Formal
  definition
                          P fe                                                  E(I(X)) = Pf
                                     N j 1              Sample mean is
                                                           an estimate



    Conventional statistical investigation                   Monte Carlo simulation
   Collect original                                    Generate sample Compute sample
                                    Fit PDF
    data sample                                           from PDF       of 0s and 1s

                                                              x1,                       I(x1),
                                                              x2,                       I(x2),
                                                                                         
                          xi                      xi          xN                        I(xN)
   0                           0
E.R.Vaidogas,VGTU, 2009                Erasmus Lecture at the University of Glamorgan            34
                          Monte Carlo method 3/4
                 Generating values of basic variables
        u       y                                             xi
                            y  FX ( x i )                                     1
                                      i                               xi  F        ( y)
                1                                                             Xi
                uj
                                                               xij

              uj+1                                          xi, j+1
               0              xi, j+1 xij            xi
                  0


                                                                    0 uj+1         uj 1   y
                          xij  FX 1 ( u j )
                                   i
                                                                                           u


E.R.Vaidogas,VGTU, 2009          Erasmus Lecture at the University of Glamorgan            35
                           Monte Carlo method 4/4
                              Methods of generating
                  Generating individual values of basic variables

  Extreme value distributions             Inverse transform method
  Exponential, Pareto-, Raleigh-,
                                          Inverse transform method
  Cauchy-distributions
  Normal distribution                     Composition method, other methods
                                                         USUALLY HIDDEN
  Gamma distribution                      Acceptance-rejection method
                                                       IN COMPUTER CODES
  Beta distribution                       Composition method, other methods
  Lognormal distribution                  Simple transformation from normal
  Truncated distributions                 Acceptance-rejection method

                          Generating vectors of basic variables
  Multi-normal distribution               Special method
  Non-normal vectors with
                                                        on transformations
                                          Methods based USUALLY HIDDEN from normal
  correlated components
                                                       IN COMPUTER CODES
                                          Multivariate transformation method,
  General case of dependence
                                          multivariate acceptance-rejection method

 For more visit, 2009
E.R.Vaidogas,VGTU,e.g.,   http://random.mat.sbg.ac.at/literature/ Glamorgan
                                     Erasmus Lecture at the University of            36
               Example: analysis of RC beam 1/7
 Failure = exceedance of the ultimate limit state in flexure

                                                        X2        X1




                                         6m

                                                   X3, X8
                                 X4


     As=33 cm2                                X7
         X6




                          X5
E.R.Vaidogas,VGTU, 2009        Erasmus Lecture at the University of Glamorgan   37
               Example: analysis of RC beam 2/7
               Limit state function & failure probability

                          g1 ( X )  M R ( X )  M E ( X )
                  M E ( X )  ( X1  X 2  X 3 X 4 X 5 ) l 2 / 8
         M R ( X )  X 7 As ( X 4  X 6  0.5( X 7 As /( X 8 X 5 ) )


Pf  P(g( X )  0) 
                         X 7 As  ( X1  X 2  X 3 X 4 X 5 ) l 2 
 P X 7 As  X 4  X 6 
                                                                
   
                        2X8 X5              8                  
                                                                   

E.R.Vaidogas,VGTU, 2009          Erasmus Lecture at the University of Glamorgan   38
               Example: analysis of RC beam 3/7
                                Basic variables

Basic                                                       General
                          Description                                         Distribution
 var.                                                       notation
  X1                Variable load                              q                   Gumbel
  X2              Permanent load                               g                  Lognormal
  X3            Density of concrete                              g               Lognormal
  X4           Cross-sectional height                            h                 Normal
  X5           Cross-sectional width                             b                 Normal
  X6           Depth to reinforcement                            a                 Normal
  X7           Yield strength of steel                           fy               Lognormal
  X8       Compressive strength of concrete                      fc                Normal




E.R.Vaidogas,VGTU, 2009          Erasmus Lecture at the University of Glamorgan           39
               Example: analysis of RC beam 4/7
             Characteristics of probability distributions

              General
Basic var.             Distribution        Mean              C.o.v.             Std.dev.
              notation
     X1          q       Gumbel          70 kN/m               0.2              14 kN/m
     X2          g     Lognormal         25 kN/m              0.07             1.75 kN/m
     X3          g    Lognormal         24 kN/m3             0.03             0.72 kN/m3
     X4          h       Normal            0.6 m              0.05               0.03 m
     X5          b       Normal           0.25 m              0.05              0.0125 m
     X6          a       Normal           0.04 m              0.07              0.0028 m
     X7          fy    Lognormal         560 MPa              0.11              61.6 MPa
     X8          fc      Normal           35 MPa              0.12              4.2 MPa




E.R.Vaidogas,VGTU, 2009       Erasmus Lecture at the University of Glamorgan               40
               Example: analysis of RC beam 5/7
                Programming the mathematical model




E.R.Vaidogas,VGTU, 2009   Erasmus Lecture at the University of Glamorgan   41
               Example: analysis of RC beam 6/7
                      Estimates of failure probability




E.R.Vaidogas,VGTU, 2009       Erasmus Lecture at the University of Glamorgan   42
                          Further examples 1/2
                          The TERECO textbook
          Chapters 1 to 3: Introduction into MC simulation + ~ 218 Examples:




E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan   43
                              Further examples 2/2
                                    Further examples
          Chapters 1 to 3: Introduction into MC simulation + ~ 218 Examples:


 Chapter 4: Loading and loading effects (27)         Chapter 11: Accumulation of damage (15)

 Chapter 5: Resistance of structural elements
                                                     Chapter 12: Serviceability (14)
 and components (18)

 Chapter 6: Safety of structural elements (26)       Chapter 13: Special situations (18)

 Chapter 7: Safety of structural components – 1st
                                                     Chapter 14: From components to systems (6)
 order theory (26)
 Chapter 8: Safety of structural components – 2nd Chapter 15: SBRA vs. Eurocodes and Multi-
 order theory (15)                                component loads (22)
 Chapter 9: Reliability of retaining walls and of    Chapter 16: Bayesian approach and other
 slopes (10)                                         updating techniques (2)
                                                     Chapter 24: Assessment of a steel frame, a
 Chapter 10: Prestressed concrete (15)
                                                     comparative study (4)


E.R.Vaidogas,VGTU, 2009                Erasmus Lecture at the University of Glamorgan             44
                          THE END OF LECTURE
                          Several concluding remarks


  An alternative approach to structural analysis &
   design
  Explicit addressing the problem of structural safety
  Monte Carlo method: computer will do everything
   instead of you
  Not a panacea, but can be helpful

                             Thanks for attention!



E.R.Vaidogas,VGTU, 2009         Erasmus Lecture at the University of Glamorgan   45

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:6
posted:8/28/2012
language:Latin
pages:45