Research Collection. Basics of structural reliability and links with structural design codes FBH Herbsttagung November 22nd, 2013.

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1 Research Collection Presentation Basics of structural reliability and links with structural design codes FBH Herbsttagung November 22nd, 2013 Author(s): Sudret, Bruno Publication Date: 2013 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Basics of structural reliability and links with structural design codes B. Sudret Chair of Risk, Safety & Uncertainty Quantification November 22nd, 2013

3 Structures and infrastructures (All images from Wikipedia) Engineers design complex systems in order to fulfill the needs of the society in terms of: Energy: nuclear power plants, dams, wind turbines, etc. Transportation: bridges, tunnels, railways, etc. Dwelling Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

4 Specifications Civil structures and infrastuctures shall be designed so as to provide services to the society and individuals while being: Safe to the public and the personnel involved in their operating Harmless to the environment Cost effective, so as to optimize the available resources Safety criteria Cost optimization Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

5 Specifications Civil structures and infrastuctures shall be designed so as to provide services to the society and individuals while being: Safe to the public and the personnel involved in their operating Harmless to the environment Cost effective, so as to optimize the available resources Safety criteria Cost optimization Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

6 Tools for structural safety How to guarantee that the designed structures are safe? Basic sciences (mathematics, computer science, statistics & probability) Engineering sciences (e.g. structural mechanics) Legal frame: norms and construction codes Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

7 Tools for structural safety How to guarantee that the designed structures are safe? Basic sciences (mathematics, computer science, statistics & probability) Engineering sciences (e.g. structural mechanics) Legal frame: norms and construction codes Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

8 Tools for structural safety How to guarantee that the designed structures are safe? Basic sciences (mathematics, computer science, statistics & probability) Engineering sciences (e.g. structural mechanics) Legal frame: norms and construction codes Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

9 Starting point: specifications on actions Service loads are defined by stakeholders w.r.t the forecast usage of the system (e.g. expected traffic on a bridge) Typical environmental actions are: Gravity Climate loads (wind, snow, temperature) Natural hazards (earthquakes) Accidental human actions (explosions) Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

10 Real world is uncertain! Aleatory uncertainty Natural hazards (earthquakes, landslides) Climate loads (wind, snow) (Variability of material properties) Epistemic uncertainty Lack of data (soils) Statistical uncertainty Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

11 How can we design structures that are safe and reliable despite uncertainties? Semi-probabilistic codes vs. Structural reliability Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

12 How can we design structures that are safe and reliable despite uncertainties? Semi-probabilistic codes vs. Structural reliability Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

13 Outline Introduction 1 Introduction 2 3 Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

14 Ingredients for classical structural design Scenarios about the usage of the structure: combinations of permanent, non-permanent and accidental loads Associated limit states Choice of design parameters: material properties (e.g. class of concrete strength) and geometry Structural model: abstract representation that allows the engineer to compute the load effects Design rules that should theoretically avoid failure when correctly applied Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

15 Semi-probabilistic design Eurocodes EC0-EC8 SIA S d (d, γ p p K ) R K γ res Resistance Computational model Characteristic value Partial safety factors Characteristic values Quantities with some statistical content, e.g. a particular quantile of the distribution Leading to pessimistic values (i.e. low resistance and high load effect) w.r.t. their nominal value Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

16 Examples of characteristic values Loads are modelled by Gumbel distributions. Eurocode 0 requires to consider 98%-quantiles: s k s 98% Resistance R is often modelled by a lognormal distribution. Eurocode 0 requires to consider 5%-quantiles: r k r 5% Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

17 Semi-probabilistic design S d (d, γ p p K ) R K γ res Resistance Computational model Characteristic value Partial safety factors Partial safety factors: A target level of reliability is fixed by the norm, which corresponds to an accepted probability of failure, e.g /year The partial safety factors are calibrated w.r.t this target reliability index They account for the intrinsic variability of the properties They also account for model errors and approximations, e.g. coming from the lack of knowledge in the physics, the numerical algorithmic errors, etc. Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

18 Conclusions Introduction Codified design is based on checking deterministic design equations These equations account for uncertainties through characteristic values and partial safety factors The partial safety factors have been calibrated with respect to a target level of reliability The true level of reliability of a particular system may be computed using the theory of structural reliability analysis Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

19 Outline Introduction 1 Introduction 2 3 Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

20 Structural reliability: problem statement Instead of a binary outcome (OK /NOK), structural reliability aims at quantifying a probability of failure Codified design Design rule S d R d Structural reliability Limit state function (failure if 0) Z = R(X R ) S(X S ) g(x) Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

21 Structural reliability: problem statement Instead of a binary outcome (OK /NOK), structural reliability aims at quantifying a probability of failure Codified design Design rule S d R d Structural model Structural reliability Limit state function (failure if 0) Z = R(X R ) S(X S ) g(x) Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

22 Structural reliability: problem statement Instead of a binary outcome (OK /NOK), structural reliability aims at quantifying a probability of failure Codified design Design rule S d R d Structural model Structural reliability Limit state function (failure if 0) Z = R(X R ) S(X S ) g(x) Characteristic values Distributions e.g. 5%-quantile for resistance Partial safety factors Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

23 Interlude: two minutes on probability theory Random variable: a variable X that can take different values in a range D X with different probability Cumulative distribution function (CDF): is the probability that X takes a value smaller than a given x F X (x) = P (X x) Probability density function (PDF): gives the probability that X is in the range [x, x + dx] f X (x) dx = P (X [x, x + dx]) f X (x) = df X(x) dx Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

24 Interlude: two minutes on probability theory Mean value (expectation): corresponds to the average outcome Variance: describes the variability around the mean value µ X = E [X] Var [X] σ 2 X = E [ (X µ X ) 2] σ X The coefficient of variation measures the amount of uncertainty: µ X CV X = σ X µ X (%) Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

25 Interlude: two minutes on probability theory Gaussian distribution A Gaussian distribution with mean value µ and standard deviation σ is defined by: PDF: CDF: where: f X (x) = 1 2πσ e ((x µ)/σ)2 /2 F X (x) P (X x) = Φ Φ(x) = x ( ) x µ σ 1 2π e t2 /2 dt Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

26 Computation of the probability of failure Simplified approach: the limit state function is the margin-to-failure Z = R S where R and S are the resistance and load effect with prescribed distributions Probability of failure Corresponds to the probability of having a negative margin P f = P (R S) = P (Z 0) From the distributions: P f = = P (R s 0) P (S = s 0) F R (s) f S (s) ds Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

27 Case of Gaussian variables Suppose the resistance R and load effect S have Gaussian distributions: R N (µ R, σ R ) S N (µ S, σ S ) Then the margin Z = R S is also Gaussian with mean µ Z /std. deviation σ Z : µ Z = µ R µ S σ Z = σ 2 R + σ2 S The probability of failure reads: ( ) 0 µz P f = Φ Φ( β) σ Z where the reliability index β reads: β = µ R µ S σ 2 R + σ 2 S Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

28 Graphical representation The reliability index β is a distance between the mean value point (µ R, µ S ) and the failure domain D f = {(r, s), r s} The most probable failure point allows to define the partial safety factors w.r.t characteristic values Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

29 General case Introduction Ingredients The design criterion cast as a limit state function g(x, M(x)) such that the failure domain is defined by the set of x with negative g-values: D f = {x : g(x, M(x)) 0} A structural model M that computes the load effects A probabilistic model of the uncertainties in the input parameters, i.e. a vector of basic random variables X of given PDF f X The probability of failure is the probability that X belongs to the failure domain: P f = P (X D f ) = f X (x) dx D f ={x: g(x,m(x)) 0} X 2 Failure domain D f = {x: g(x) 0} Safe domain Ds X 1 Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

30 Computational methods Monte Carlo simulation Principle Assess the design of a large number n of virtual structures whose parameters are drawn according to the assumed variability of the load/resistance. Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

31 Computational methods Monte Carlo simulation Principle Assess the design of a large number n of virtual structures whose parameters are drawn according to the assumed variability of the load/resistance. Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

32 Computational methods Monte Carlo simulation Principle Assess the design of a large number n of virtual structures whose parameters are drawn according to the assumed variability of the load/resistance. Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

33 Computational methods Monte Carlo simulation Principle Assess the design of a large number n of virtual structures whose parameters are drawn according to the assumed variability of the load/resistance. Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

34 Computational methods Monte Carlo simulation Principle Assess the design of a large number n of virtual structures whose parameters are drawn according to the assumed variability of the load/resistance. Probability of failure ˆP f = n f n = # failed systems # virtual systems Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

35 Advanced computational methods All structural reliability methods are based on repeated evaluations of the limit state function, which depends on the structural model Monte Carlo simulation is not compatible with complex (e.g. finite element) models FORM/SORM Based on the computation of a most probable failure point P Used to establish partial safety factors, e.g. when considering non Gaussian variables, or for updating the reliability (SIA 269) Surrogate models Response surfaces (a.k.a surrogate models) may be built using advanced statistical representations (polynomial chaos expansions, Kriging) They are used for structural reliability analysis but also for parametric sensitivity analysis or optimization Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25

36 Conclusions Introduction is the backbone of the modern structural design codes The machinery of partial safety factors is established based on this probabilistic reasoning and allows for fast design for common structures When dealing with uncommon structures / reassessment of existing structures / high levels of reliability, the engineer should resort to a full probabilistic analysis Thank you very much for your attention! Bruno Sudret (Chair of Risk & Safety) FBH Herbsttagung November 22nd, / 25