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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

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