A probabilistic approach to design civil engineering structures Gabrielle Muller & Gabriele Albertini Semester project Spring term 2015 Professor Jean-Franc¸ois Molinari Supervisor Fabian Barras Computational Solid Mechanics Laboratory -LSMS Ecole Polytechnique Fed´ erale´ de Lausanne -EPFL CONTENTS Introduction 1 I Theoretical considerations and principles of probabilistic approach in civil engineering2 1 about probabilistic and deterministic approach in struc- turedesign 3 1.1 Basic structural design principles 3 1.2 The deterministic approach - The classical approach in civil engi- neering 3 1.3 The probabilistic approach 6 2 mathematical and numerical tools required for a proba- bilistic approach 11 2.1 Define Probability Density Functions and the Monte Carlo method 11 2.2 The Monte Carlo Method 14 2.3 Advantages and limitations of using Monte Carlo 16 II The application of probabilistic theory: from a basic ex- ample in Python to a real structure in Akantu 17 3 anillustrativeexampleofthemontecarlotheoryinpython 18 3.1 Evolution of the accuracy of the results 18 3.2 Study of the accuracy of the Monte Carlo method 23 4 an illustrative example of the monte carlo theory using akantu 28 4.1 Getting started in Akantu 28 4.2 The Model 28 4.3 The analytical method 29 4.4 The Monte Carlo Simulation 29 4.5 Results 29 ii CONTENTS 5 using akantu for an application of the monte carlo the- ory 32 5.1 Introduction 32 5.2 Considerations about the model 33 5.3 The Monte Carlo simulations 39 5.4 Results 40 Conclusion 50 iii INTRODUCTION The goal of this semester project is to study a probabilistic approach to design civil engineering structures. In fact, currently most structural design is done via a deterministic approach, being the easiest, fastest and best-known approach to engineers. Even if deterministic design guarantees a certain structural safety pro- posed by the SIA standard prescriptions, it is of interest to consider a probabilistic approach in order to quantify structural safety and reliability that cannot be as- sessed with a deterministic approach. The differences between deterministic and probabilistic design approached will be studied and consequences for future re- lated work in structural engineering will be drawn. • This report is divided in two parts: – Part 1 is the theoretical part which exposes the concepts of determinis- tic design and probabilistic design in structural engineering and intro- duces the Monte Carlo theory. – Part 2 is the application of the theory and consists in an implemen- tation of the Monte Carlo theory to real examples. Starting from the simple case of a steel bar subjected to axial loading, using open-source software Akantu, we implemented the Monte Carlo theory to assess reliability of an existing dam. The goal was to include probabilistic theory in the frame of dam design principles. • Numerical tools used during this project Throughout this project, we used the Akantu software through a Python interface. Akantu is an open source Finite Element library implemented at the Laboratory of Computational Solid Mechanics (LSMS) at EPFL. Gmsh was used to create the mesh for the dam. • Supervision This project was done within the LSMS at EPFL under the direction of Professor Molinari. The supervision was done by Fabian Barras, who we collaborated closely with throughout the term. 1 Part I. Theoretical considerations and principles of probabilistic approach in civil engineering 2 1 ABOUTPROBABILISTICANDDETERMINISTICAPPROACH INSTRUCTUREDESIGN 1.1 basic structural design principles Most engineering problems can be solved by the confrontation of the two follow- ing quantities: • solicitation or stress S S can be the resulting bending moment in a beam subjected to well-defined loads, deflection of a beam under a given load case or ground solicitation due to external loads. • corresponding capacity or resistance R R can be the ultimate resistant moment of a beam, the maximum allowable deflection for a beam or the shear capacity. Structural safety requires that R > S. Failure occurs whenever R < S. This gen- eral formulation is applicable to most civil engineering problems. (These considerations are directly taken from course support [1]) 1.2 the deterministic approach - the classical approach in civil engineering Currently, deterministic approach is the method most widely used by civil engi- neers when designing structures. Design of structural elements is done as follows: the design value of resistance Rd is compared to the corresponding solicitation value Sd. Structural safety re- quires that Rd > Sd (1) 3 1.2 the deterministic approach - the classical approach in civil engineering The design value Rd is the resistance value R divided by a safety coefficient gR that takes into account simplifications linked to the model, uncertainties and variability of the material properties. The design value Sd takes into account the different scenarios taken into ac- count as well as the corresponding loads. Here again, a safety coefficient gS is defined in order to include simplifications. When designing a building, one has to differentiate between predominant and concomitant effects, inducing different safety factors. However, this differentiation is not relevant in the context of this project and will not be done for the rest of this report. The most characteristic feature of the deterministic approach is the fact both resistance and solicitation are defined by a fixed value, the characteristic value, being the result of multiple considerations and scenarios. In the concept of this fixed value lies the major difference with the probabilistic approach [1]. As a help for structural engineers, the Eurocode and the SIA standards give guidelines about how determining a characteristic value for an action as well as the safety coefficients. The guiding principles of the SIA standards are described in section . 1.2.1 SIA Standard prescriptions for deterministic design • design value of solicitation – characteristic value Sk ∗ generally, the characteristic value corresponds to a the probability of non-exceedance of pt = 1 − p f = 95% in the case of a normal law. This means that there is a probability of 5% for the Sk to be exceeded. – safety factor gS ∗ this factor takes into account the lack of precision of the character- istic load value with the factor g f and the uncertainties linked to the structural analysis gm. Values for both factors are well defined in the Standard. In the case of linear relationship between actions and action effects, the safety factor gS is computed as gS = g f · gm (2) 4 1.2 the deterministic approach - the classical approach in civil engineering ) the design value for solicitation is then defined by equation 3 Sk Sd = (3) gS • design value of resistance is a function of the design values of – material properties Xd – geometric data ad – uncertainties due to the resistance model are taken into account with the safety factor gR and the material properties with the factor gm. gM = gR · gm ) the design value for resistance is defined in 4 Rk Rd = (4) gM The considerations of this entire section are mostly taken from course [1]. 1.2.2 Considerations in relation to the 5% fracture percentile It is interesting to notice that the deterministic design approach is nevertheless based on probabilistic design. Indeed, the characteristic values are determined through a fixed percentile via the normal law distribution (in most cases, but it could theoretically also be another probabilistic law). With respect to this, it is important to mention that the 5% fracile is not a fixed percentile, but depend on the level of reliability that wants to be achieved. The higher the required level, the smaller the resulting fractile and thus the bigger the characteristic value. Moreover, the safety factors given in the Standard for the deterministic approach hide a probabilistic procedure which they were determined with. The methodol- ogy for the determination of these safety factors will be explained in section1.3. ) This deterministic procedure given by the SIA Standards gives reliability to the design and confidence to the engineer in charge of the structural design, but this reliability is not quantified. It only says if yes or no the studied element passes the verification for a given risk scenario, but in any cases it does not give a quantitative sense of how far the structure is close to failure. The deterministic approach may look secure to most engineers and make them pretend to design 5 1.3 the probabilistic approach safe structures, but in reality it does not give a good understanding of the reliabil- ity of the structure. The probabilistic method exposed in 1.3 balances this lack of quantification of reliability and suggests a way to evaluate reliability of structures. According to the different possible verification formats, how can reasonable safety be defined considering uncertainties related to different parameters? Chapter 7,[1] An attempt to answer this question is done in section 1.3. 1.3 the probabilistic approach The major advantage of designing structures with a probabilistic approach is the possibility to quantify the reliability of the structure. Instead of using character- istic values which correspond to upper or lower boundary values, a probabilistic approach allows engineers to quantify the reliability of the designed structures, as opposed to deterministic design which only allows to determine whether yes or no the structure is safe. In most cases, the probabilistic approach of designing a structure gives results that are closer to reality and thus less conservative than a deterministic approach. This could be of interest in structural design since it would allow to design structures differently and save on materials and on money, as well as assessing the reliability of an existing structure and determining how far it is from failure. Moreover, it is an useful tool for assessing the reliability of existing structures since parameters can be adapted with respect to target reliabil- ity or importance of the building.