Construction Reliability: Safety, Variability and Sustainability

Preface

From 26 to 28 March 2008, a conference entitled Fiabilité des matériaux et des structures (Reliability of materials and structures, JNFiab’08)¹ took place at the University of Nantes, France, bringing together the French scientific communities interested in reliability and risk analysis, as applied to materials and structures. This colloquium followed on from several different events: the fifth Reliability of materials and structures conference, the second Méc@proba training day² and the second scientific session in the subject area of Understanding risk in civil engineering (MRGenCi scientific interest group³).

It combined their themes and concerns as an extension of the first shared workshop between the Associations Françaises de Génie Civil (AFGC, or French Associations of Civil Engineering⁴) and the Associations Françaises de Méchanique (AFM, or French Associations of Mechanical Engineering)⁵, during the twenty-fifth annual meeting of the Association Universitaire de Génie Civil (AUGC, Universities civil engineering association⁶) held on 23–25 May 2007, in Bordeaux, France.

This book was first conceived during these sessions, organized by the MRGenCi and Méc@Proba scientific interest groups, where the authors gave presentations on the advances they have made in their respective fields.

Although the examples of structures that can be found in this book fall under the umbrella of civil engineering (nuclear and oil industries, buildings and dams), themethods we consider are just as applicable to any sort of complex mechanical system involving a large number of uncertainties. Thus the book is of interest to the civil engineering community but also to mechanical engineers or those interested in reliability theory, whether their background is in industry or academia, who have been exposed to research and development processes. Masters students, engineering students and doctoral students, engineers and research associates will all find a detailed discussion of methods and applications.

The authors are indebted to the two main proofreaders, with their complementary backgrounds. The first is Maurice Lemaire, a university professor who teaches at the Institut Français de Mécanique Avancée (IFMA, French Institute for Advanced Mechanical Engineering⁷) and at the Blaise Pascal University⁸ (UBP) at Clermont-Ferrand, and who is consultant to the company Phimeca⁹ which he co-founded. The second is André Lannoy, Vice-President of the Institut pour la Maîtrise des Risques (IMdR, Institute for Risk Management¹⁰), who built his career as a research engineer and subsequently as scientific adviser to the research and development section of EDF. In particular, André Lannoy co-organizes the working group Sécurité et sûreté des structures (GTR 3S, Safety and reliability of structures¹¹), a group which counts several of the authors of this book among its members.

The authors would like to express their particular gratitude to Maurice Lemaire for his contributions to the development of the field of structural reliability. The few brief paragraphs below give a short overview of his career and his contributions to scientific production, advocacy and above all to training, which has inspired several of the authors to pursue their career paths.

After receiving his diploma from the Institut National des Sciences Appliquées, (INSA¹², National Institute for Applied Sciences), in Lyon in 1968, followed by further studies in applied mathematics (1969), Maurice Lemaire received his doctorate in engineering (1971). Following his higher state doctorate (1975), he was appointed a position at CUST (the engineering school that became the Polytech’ Clermont-Ferrand¹³), within the Blaise Pascal University (1976). He was involved in the founding of the IFMA (1987), where he has been a professor since 1991.

Maurice Lemaire founded the Laboratoire de Recherches et applications en Mécanique Avancée (LaRAMA, Research and applications in advanced mechanics research group) in 1989 (and the IFMA/UBP laboratory, which is now LaMI¹⁴), where he has been a director since 2005. Head of research at IFMA from 1991 to 2007, he has supervised 44 doctoral students and participated in 180 thesis examinations.

Co-founder of the Fiabilité des matériaux et des structures symposium¹⁵ (Cachan, 1994), co-organizer of the 7th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP¹⁶, 1995 in Paris), and then president of the International Civil Engineering Risk and Reliability Association (CERRA¹⁷) from 1995 to 1999, he is a member of the scientific committee of the International Federation for Information Processing’s working group 7.5 (IFIP WG 7.5) on the reliability and optimization of structural systems¹⁸. Maurice Lemaire is also a founder of the Méc@proba meetings¹⁹ (Marne-la-Vallée, France, 2006).

He is a promoter for the scientific commission Mécanique probabiliste des matériaux et des structures (Probabilistic mechanics of materials and structures²⁰) of the AFM), and is responsible for numerous industrial research projects. He also contributes scientific insight and ongoing supervision into the growth of the company Phimeca Engineering, which he co-founded in 2001.

There can be no doubt that Maurice Lemaire has contributed to the development of the science of structural reliability at a national and international level.

The authors therefore recommend that readers consult his publications, which are cited throughout this book.

The authors would like to express their gratitude towards the MRGenCi scientific interest group which made it possible to write this book. That group, created on the initiative of Messrs Breysse (Professeur at University of Bordeaux I), Boissier (Professeur at UBP), Melacca (SMA-BTP) and Gérard (PDG OXAND SA), now consists of more than 30 industrial companies, learned societies, universities and engineering schools.

We cannot speak highly enough of the authors, proofreaders and all the other people who have contributed to this book and to ensuring that it is received by a wide audience.

Julien BAROTH

Franck SCHOEFS

Denys BREYSSE

JUNE 2011

1 http://www.sciences.univ-nantes.fr/jfms2008/DownloadJFMS2008.pdf.

2 http://www.lamsid.cnrs-bellevue.fr/vf/actualites/journee_proba/plaquette_proba.pdf

3 http://www.mrgenci.u-bordeaux1.fr/.

4 http://www.afgc.asso.fr/.

5 http://www.afm.asso.fr/.

6 http://www.augc.asso.fr/.

7 http://www.ifma.fr/.

8 http://www.univ-bpclermont.fr/.

9 http://www.phimeca.com/.

10 http://www.imdr.eu.

11 http://www.imdr.eu/v2/extranet/detail_gtr.php?id=36.

12 http://www.insa-france.fr/.

13 http://www.polytech-clermontferrand.fr/.

14 www.ifma.fr/lami/.

15 www.sciences.univ-nantes.fr/jfms2008/DownloadJFMS2008.pdf

16 www.icassp2011.com/.

17 www.ce.berkeley.edu/projects/cerra/.

18 www.era.bv.tum.de/IFIP/.

19 www.lamsid.cnrs-bellevue.fr/vf/actualites/journee_proba/plaquette_proba.pdf

20 www.afm.asso.fr/PrésentationdelAFM/Commissions/MPMS.

Introduction ¹

Background and objectives

This book describes and illustrates methods to improve prediction of the lifetime and management of civil engineering structures. This contributed collection aims to complete the existing literature and to provide access to both recent scientific approaches and examples of applications in study cases. The authors are drawn from amongst university academics, senior engineers and scientific managers in companies. Amongst others, Ditlevsen & Madsen [DIT 96], Melchers [MEL 99] and Lemaire [LEM 09] have already made significant contributions to structural reliability, that is to say, the study of the ability of a structure to perform a function (depending on its environment, life, etc.). Favre [FAV 04], [SUD 08b], by contrast, studied geotechnical structures. These last books introduced concepts of uncertainty related to materials and loads, as well as statistical and probabilistic methods applied to civil engineering. This book also uses these basic concepts, and notions of uncertainty or reliability, in particular, are discussed in this introduction.

The authors’ main objective in this book is the presentation of recent methods of data processing and computation which have not been widely disseminated. Many of these methods have already been used in industrial applications: this book aims to present these applications through case studies and examples over a wide set of materials, buildings and structures in civil engineering.

If the examples of structures presented in the following pages are limited to the field of civil engineering (whether in nuclear, oil, or dam building applications), the methods presented are applicable to any complex mechanical system in an uncertain environment. This book is mainly intended for those familiar with civil engineering, engineering mechanics or the theory of reliability, both from industry and academia, involved in research and development. Students, especially engineering students and PhD students, engineers and research fellows may also have an interest in the book.

A civil engineering project is considered here to be a system that ensures one or more global functions. It consists of components, sub-structures, structures, human actors, procedures, and organization in a given environment. It performs the basic functions that contribute to the achievement of these global functions. The system is identifiable, and can be broken down into the functions it performs (a functional approach), or into structurally interdependent elements or subsystems (a structural approach). A civil engineering project is considered safe when it is fit for duty for the duration of its service life, without damage to itself or to its environment (French norm X60-010). The safety of a structure can be quantified by its probability of not providing any of its functions, at any moment of its expected lifetime.

This probability of failure, denoted Pf, is mathematically complementary to the system reliability, often denoted R = 1-Pf.

The objective of this book is to present the methods of functional analysis (Parts 1, 4 and 5) and those of structural analysis (throughout the entire book), when the systems studied are in a working condition that does not induce damage to people, property or the environment. These systems are then considered to be safe. A significant section of the book is devoted to the calculation of reliability, which may or may not be dependent on time (see Parts 3 and 4).

Some of the concepts developed throughout the book are introduced in the following section, which also raises a number of questions that the book later tries to answer and illustrate.

Qualitative and quantitative methods of safety assessment for a construction project

The study of the safety of a civil engineering structure may be conducted by system analysis, followed by an analysis of failure modes and modeling of failure scenarios, in order to answer some crucial questions:

Question 1. How can we highlight the most likely failures and the most critical failure scenarios, which could potentially be the basis of risk analysis?

Question 2. What are the preventive measures that can improve system safety?

Part 1 of this book addresses these issues. It presents some methods of qualitative assessment of structural safety. These methods allow us to analyze a system, its failure modes, and to model the failure scenarios in order to evaluate their criticality. An application of methods to assess the criticality of scenarios is then proposed for a hydro civil engineering work.

Moving beyond qualitative methods, quantitative methods are also presented in Parts 2 to 5.

For a given failure scenario, a third question might be:

Question 3. How can we evaluate and reduce, when possible, the probability of events causing failure or, again when possible, how can we evaluate and reduce the gravity of failure consequences?

Some answers to this question are given by an analysis of uncertainties in knowledge of a structure and the actions applied to it. It has been a deliberate choice in this book to focus on early uncertainties, in particular focusing on uncertainties in our knowledge of building materials. With regard to the study of applied actions, readers can refer to the following sources: [CEN 03], [CEN 05] on snow and wind climate loads, [SCH 08], [SMI 96] for wave action, or [PEY 09] regarding loads from hydrology.

Materials with uncertain properties

There are many sources of uncertainty related to materials. Being aware of them, or reducing them, will achieve greater safety [FAV 04]. Thus, the whole book attempts to answer the following questions:

Question 4. How can we represent and use uncertain data, describing the geotechnical characteristics of materials? What are the consequences of heterogeneity and variability for structural safety?

A first classification consists of combining uncertainties into two categories concerning the condition of the work considered:

– uncertainties related to internal variables affecting the internal condition of the structure, such as material properties (elastic modulus, Poisson’s ratio, density, coefficient of thermal expansion, etc.), geometrical parameters (dimensions, moments of inertia, etc.), internal boundary conditions (excluding actions on areas or volumes) and internal links;

– uncertainties related to external variables beyond the internal condition of the structure, such as natural actions (wind, waves, snow, temperature, earthquake, etc.) and operating loads (loads, operating loads, etc.).

In some cases these differences may be small, e.g. for a metal sheet pile wall, where uncertainty about the weight of the soil near the wall is external if we consider the resistance of the curtain, and internal if we consider stability (sliding). Table I.1 provides a more detailed typology of uncertainties that are not problematic.

Table I.1. Typology of uncertainties and means of action

A second classification distinguishes random (or intrinsic) uncertainties and epistemic uncertainties, which are sometimes measurable and are likely to be modeled by random variables. This probabilistic format is easier to introduce into behavior models and to propagate through structural analyses. Amongst the uncertainties, we encounter the physical uncertainties that are inherent to the nature of a system, such as uncertain parameters: strength, loads, environment, geometric characteristics, boundary conditions, etc. However, uncertainties due to lack of knowledge of physical phenomena are epistemic because they can be reduced by getting more information, or by doing further research.

However, the gap between a real system and a model, whether simple or complicated, can also lead to uncertainty. This type of model uncertainty is epistemic: it can sometimes be incorporated into a calculation of reliability through the introduction of additional random variables to represent the dispersion of the model results (analytical, numerical or experimental) compared to the physical phenomenon.

Table I.1 presents more ontological (or phenomenological) uncertainties, harder to identify than others. It describes the uncertainties attached to random processes or systems which have not been imagined (for example, the Concorde crash in Paris in 2000) or which are too complex to be modeled [TAN 07].

Engineers should adopt an alternative attitude, just as philosophers should avoid the question: is reality random or not? Engineers are simply interested in objects, processes and phenomena that are not entirely predictable: uncertainty concerns engineers to the extent that it limits their ability to forecast [MAT 08]. More than the classification of uncertainties (which depends on a refinement of the models used), the priority is how to reduce or control them (as raised in Questions 3 and 4).

Eurocode principles

The European building code (Eurocodes) has been developed over the last 20 years. This code allows the variability of materials in a deterministic formalism to be better accounted for, based on values defined on a probabilistic basis. This formalism is also based on failure situations formalized according to their consequences in terms of Serviceability Limit States (SLS) and Ultimate Limit States (ULS). SLS correspond to conditions beyond which the serviceability requirements specified for the work (or a part of it) are not met. These statements are generally related to demands of comfort (vibration, deflection), aesthetics (excessive distortion), durability (cracking, corrosion), the proper functioning of equipment (insulation, sealing, etc.), without resulting in short-term collapse of a structure. ULS are associated with the collapse of a structure, or other forms of structural failure: loosing the equilibrium of the work, reaching the maximum strength of a part of the structure, failure of the subgrade; it is defined as the accidental limit state (or progressive failure) for systems exposed to variable loads and to a fatigue limit state (condition of sustainability). The principle of the Eurocodes can be summarized as:

– proposing an appropriate limit state for each failure scenario;

– characterizing the parameters affecting the considered limit state criteria, using probabilistic and statistical tools;

– replacing the complete distributions of the probabilistic variables by average values and dispersion or, by characteristics values;

– neglecting the dispersion of some data, considering them as deterministic;

– taking into account the other neglected uncertainties, by introducing fixed coefficients.

This regulatory approach is characterized as semi-probabilistic, because it incorporates a probabilistic modeling of uncertain parameters (b), while introducing partial safety factors (e). The second part of the Code recalls the definitions of particular values (characteristics values), (c), and uses the concept of semi-probabilistic design. The Eurocodes are a compromise between inadequate deterministic design, and impractical probabilistic design.

We can also make a distinction between four levels of reliability methods:

– Level I: methods using some specific values for the variables (resistance and load, for example); each random variable is represented by a characteristic value with statistical content which is often poorly defined;

– Level II: methods devoted to characterizing the random variables by their mean and variance;

– Level III: methods requiring more knowledge of the joint probability distribution of all the random variables, and allowing a reliability index and failure probability to be obtained as a measure of safety;

– Level IV: methods intending to provide a level of reliability integrating economic criteria: for example, taking into account the costs of construction, maintenance, repairs, failure consequences, etc.

One or other of these four levels of reliability methods may be considered, depending on the size of the problem to be analyzed.

For example, Level IV methods are devoted to the analysis of nuclear power plants, whilst Level I methods are applied when studying simple structures with lower stakes (warehouse storage products that are safe, etc.). The reliability assessment of structures with high stakes is theoretically Level III. However, a simplification is necessary, because the joint probability distribution of variables is difficult to assess. We usually just know the marginal distributions, sometimes with correlation coefficients to model the interdependence between variables. Such methods are called advanced Level II methods.

Variability and heterogeneity of materials

Part 2 of this book seeks to answer Question 3 by showing how to use available data to describe their heterogeneity and variability. Chapter 4 deals with the characterization of uncertainty in geotechnical data. This part provides a complete set of methods: the identification of sources of uncertainty described above, the classification of data (outliers, censored and poor data), and statistical representation and modeling of these data (possibilistic or probabilistic). Readers are referred to other publications on this topic including [CEL 06], [LEM 99], where answers to Question 3, and to Question 5 (below) can be found.

Question 5. How can we use limited or censored data?

Chapter 5 presents estimates related to material variability (average, characteristic values), introduces geostatistical modeling tools (variograms, estimation and simulation methods) [MAR 09]. Chapter 6 provides an example of a shallow footing for which reliability is considered; the effect of soil variability on the variability of the bearing capacity and safety of the footing is studied; finally, the spatial correlation is taken into account to study its influence on the safety of the linear footing.

Part 4, devoted to problems of time-variant reliability, completes Part 2 by showing how the enrichment of statistical analysis and the use of Bayesian approaches can be applied to samples of a small size (Chapter 11).

The computation of reliability-coupled mechanical and reliability models

Parts 2 to 5 of the book use the calculation of reliability indicators and answer the question:

Question 6. How can we quantify the reliability of a system or a structure?

This calculation involves various levels of complexity, concerning mechanical and probabilistic analyses. The complexity of the probabilistic model depends on the distributions of the variables, their physical limitations and their interdependence. The complexity of the mechanical behavior stems from its size (number of components), its transients and nonlinearities, etc.

When an operating system depends on its mechanical condition, there is an interdependence between the roles of mechanical and reliability variables; this is called mechanical-reliability coupling [LEM 00], which can be described in five stages:

– identify the purpose of the structure: its function, behavior, operating conditions and possible failures (resulting from failure analysis);

– develop predictive models of mechanical behavior with and without defects, and probability distributions of the design variables;

– identify the possible failure scenarios. The appropriate functioning of the system is defined by the performance function Gi (or limit state) to be complied with. Failure is reached if one of these limit-states is exceeded. This analysis, often overlooked, is crucial and must be as thorough as that undertaken in (b);

– for each failure scenario, determine the reliability level and the sensitivity factors. These latter are very useful in decision making for quality control and system optimization; finally,

– assess the overall failure probability of the structural system and define the partial factors to be used for the calibration of the codes and regulations.

We take x to be the vector of a model’s uncertain parameters xi , e.g. external actions (load, wind, wave, earthquake) or geometrical characteristics (size, area and moment of inertia of cross-sections), material properties (yield stress, Young’s modulus, Poisson’s ratio, etc.). We model each parameter with a random variable Xi , characterized by a probability distribution representing the uncertainty related to this parameter (probabilistic model). This can be done using statistical studies, physical observations, or expert advice (usually a possibilistic model in that case).

Each failure scenario is associated with a performance function (also known as a limit state function, or a safety margin), denoted G. The inequality G > 0 indicates the safety domain Ds, and inversely G ≤ 0 indicates the failure domain Df. The objective is to evaluate the probability Pf that the realizations of random variables belong to the failure domain. In the simple case of two variables representing the resistance R and load S, the performance function (or safety margin) is written as G (R, S) = R-S. In practice, the statistical parameters of the stress S, and sometimes those of the resistance R, are not directly accessible, because the measurements and observations are made only for the basic uncertain parameters of the vector x from which R and S originate. The variable G and the random vector X modeling x are connected by a mechanical transformation: G(R, S) = G(X).

We can consider that this transformation is known (e.g. calculating a loading effect from a given depth of snow), although in some cases it is only available using an algorithm, as for example, using finite element software. In this book, the reliability indicators most often used are failure probability and reliability indices (both Cornell and Hasofer–Lind; see [LEM 09] for a detailed presentation):

– the probability of failure is evaluated by integrating the joint density function over the failure domain Df, where:

– the Cornell index is the distance between the median point of the G margin and the point where the margin becomes zero (i.e. the failure point); this distance is measured in terms of number of standard deviations. In other words, if mG and σG represent the mean and the standard deviation of G, respectively, the Cornell index βC is written as: βC = mG / σG. If the margin is normally distributed, we can easily show that the failure probability is:

where Φ is the standard normal cumulated distribution function. This expression is accurate when the distribution of G is Gaussian (a linear combination of two Gaussian variables, such as R-S, is Gaussian). If this condition is not verified, the Cornell index only gives a measurement that is no longer explicitly linked to the probability of failure. It is even less useful, as it implies assumptions such as the linearity of the margin and the normality of distributions;

– the Hasofer–Lind reliability index [HAS 74] is an invariant estimator of reliability. Hasofer & Lind proposed it to change physical variables into standard Gaussian variables (i.e. with zero means and unit of standard deviation) and so that they would be statistically independent. In this so-called standard space, the failure probability is written according to the failure domain H ≤ 0 so that:

where φnis the probability density function of the n-variate standard normal distribution. According of Hasofer–Lind’s definition, the reliability index β is the minimum distance between the origin and the limit state surface in the standard space. This distance is deduced from a hyperplane tangent to the limit state function at a point P*, called the design point. Finding β is thus just a question of solving the following optimization problem:

Reliability methods

Different approaches are possible for coupling mechanical and reliability methods, each offering various levels of compromise between accuracy, cost and reliability indicators, with regard to the range of validity of the methods (strongly or weakly nonlinear (y=x^1.1 is weakly nonlinear, y=x^5 is strongly nonlinear) mechanical calculations, or a more or less reduced number of random variables, etc.).

Two classes of conventional methods cover the majority of current developments and applications: the Monte Carlo method and First/Second Order Reliability Methods (FORM/SORM approximations). These are introduced and used in Parts 2 to 5.

Monte Carlo simulations are the most robust method to assess the failure probability of a complex system. They enable the achievement of reference results and control other types of approximation. However, they are often expensive. In general, to properly assess probability of the order of 10-n, we must perform 10n+² to 10n+³ mechanical calculations. It is obvious that this method is impossible to use for large systems with a low probability of failure. More efficient techniques such as modified Latin Hypercube, used in [SCH 08], can be considered as alternatives.

In a standard space, the FORM/SORM methods are based on the evaluation of a reliability index, denoted β , followed by an approximation of the probability of failure. The Hasofer–Lind Index [HAS 74] is the most commonly used. The search for the design point P introduced earlier can be performed by an optimization method appropriately chosen with respect to the particular form of the problem.

A first approximation of Pf is obtained by replacing the boundary condition H(ui ) = 0 by a tangent hyperplane at the design point; this approximation is known as the First Order Reliability Method (FORM). Taking into account the rotational symmetry property of the standard probability density, we can estimate this probability by [DIT 96]:

where Φ is the standard cumulative distribution function. The accuracy of this approximation depends on the nonlinearity of the limit state, especially in the vicinity of P*.

The purpose of Part 3 of the book is to present another class of methods for calculating reliability, called response surfaces, since the mechanical response, usually not explicit, is approximated by a meta-model, which is often reduced to an explicit analytical polynomial function. This group of methods has been the subject of recent developments, which are applied to examples of a truss, then to the skeleton of a building on several floors. Therefore, Part