a probabilistic approach to design civil engineering ......1.3 the probabilistic approach 6 2...

A probabilistic approach to designcivil engineering structures

Gabrielle Muller & Gabriele Albertini

Semester projectSpring term 2015

Professor Jean-Francois Molinari

SupervisorFabian Barras

Computational Solid Mechanics Laboratory -LSMSEcole Polytechnique Federale de Lausanne -EPFL

C O N T E N T S

Introduction 1

I Theoretical considerations and principles of probabilisticapproach in civil engineering 2

1 about probabilistic and deterministic approach in struc-ture design 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 an illustrative example of the monte carlo theory in python 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

I N T R O D U C T I O N

The goal of this semester project is to study a probabilistic approach to designcivil engineering structures. In fact, currently most structural design is done viaa deterministic approach, being the easiest, fastest and best-known approach toengineers. Even if deterministic design guarantees a certain structural safety pro-posed by the SIA standard prescriptions, it is of interest to consider a probabilisticapproach in order to quantify structural safety and reliability that cannot be as-sessed with a deterministic approach. The differences between deterministic andprobabilistic 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 thesimple case of a steel bar subjected to axial loading, using open-sourcesoftware Akantu, we implemented the Monte Carlo theory to assessreliability of an existing dam. The goal was to include probabilistictheory in the frame of dam design principles. .

• Numerical tools used during this projectThroughout this project, we used the Akantu software through a Pythoninterface. Akantu is an open source Finite Element library implemented atthe 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 thedirection of Professor Molinari. The supervision was done by Fabian Barras,who we collaborated closely with throughout the term.

1

Part I.

Theoretical considerations andprinciples of probabilistic approach in

civil engineering

2

1

A B O U T P R O B A B I L I S T I C A N D D E T E R M I N I S T I C A P P R O A C HI N S T R U C T U R E D E S I G N

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-definedloads, deflection of a beam under a given load case or ground solicitationdue to external loads.

• corresponding capacity or resistance RR can be the ultimate resistant moment of a beam, the maximum allowabledeflection 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 resistanceRd 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 coefficientγR that takes into account simplifications linked to the model, uncertainties andvariability 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 γS isdefined in order to include simplifications. When designing a building, one hasto differentiate between predominant and concomitant effects, inducing differentsafety factors. However, this differentiation is not relevant in the context of thisproject and will not be done for the rest of this report.

The most characteristic feature of the deterministic approach is the fact bothresistance and solicitation are defined by a fixed value, the characteristic value,being the result of multiple considerations and scenarios. In the concept of thisfixed value lies the major difference with the probabilistic approach [1].

As a help for structural engineers, the Eurocode and the SIA standards giveguidelines about how determining a characteristic value for an action as well as thesafety coefficients. The guiding principles of the SIA standards are described insection .

1.2.1 SIA Standard prescriptions for deterministic design

• design value of solicitation

– characteristic value Sk

∗ generally, the characteristic value corresponds to a the probabilityof non-exceedance of pt = 1− p f = 95% in the case of a normallaw. This means that there is a probability of 5% for the Sk to beexceeded.

– safety factor γS

∗ this factor takes into account the lack of precision of the character-istic load value with the factor γ f and the uncertainties linked tothe structural analysis γm. Values for both factors are well definedin the Standard. In the case of linear relationship between actionsand action effects, the safety factor γS is computed as

γS = γ f · γm (2)

4

1.2 the deterministic approach - the classical approach in civil engineering

⇒ the design value for solicitation is then defined by equation 3

Sd =SkγS

(3)

• design value of resistanceis a function of the design values of

– material properties Xd

– geometric data ad

– uncertainties due to the resistance model are taken into account withthe safety factor γR and the material properties with the factor γm.

γM = γR · γm

⇒ the design value for resistance is defined in 4

Rd =RkγM

(4)

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 neverthelessbased on probabilistic design. Indeed, the characteristic values are determinedthrough a fixed percentile via the normal law distribution (in most cases, but itcould theoretically also be another probabilistic law). With respect to this, it isimportant to mention that the 5% fracile is not a fixed percentile, but depend onthe level of reliability that wants to be achieved. The higher the required level, thesmaller the resulting fractile and thus the bigger the characteristic value.Moreover, the safety factors given in the Standard for the deterministic approachhide 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 tothe design and confidence to the engineer in charge of the structural design, butthis reliability is not quantified. It only says if yes or no the studied elementpasses the verification for a given risk scenario, but in any cases it does not givea quantitative sense of how far the structure is close to failure. The deterministicapproach 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 ofquantification of reliability and suggests a way to evaluate reliability of structures.

According to the different possible verification formats, how can reasonable safety bedefined considering uncertainties related to different parameters? Chapter 7,[1]

An attempt to answer this question is done in section 1.3.


The major advantage of designing structures with a probabilistic approach is thepossibility to quantify the reliability of the structure. Instead of using character-istic values which correspond to upper or lower boundary values, a probabilisticapproach allows engineers to quantify the reliability of the designed structures,as opposed to deterministic design which only allows to determine whether yesor no the structure is safe. In most cases, the probabilistic approach of designinga structure gives results that are closer to reality and thus less conservative thana deterministic approach. This could be of interest in structural design since itwould allow to design structures differently and save on materials and on money,as well as assessing the reliability of an existing structure and determining howfar it is from failure. Moreover, it is an useful tool for assessing the reliability ofexisting structures since parameters can be adapted with respect to target reliabil-ity or importance of the building.

Some basic concepts related to probability and statistics need to be reminded:

• probability density, with the mean value and standard deviation

• distribution function, with median value

• normal distribution, widely used for engineering and science applications

In the context of structural design in civil engineering, it is of interest to quantifythe safety and reliability of a structure, especially for existing structures.

1.3.1 Quantifying the reliability of a structure

A relevant approach of quantifying the reliability of a structure is to estimate theprobability of failure. Probability of failure is a reliable indicator of structuralsafety and a useful tool from an engineering point of view.

6


The basic principles of statistics and probability reminded in section1.3 can beapplied to a probabilistic analysis of structural safety and allow to mathematicallyexpress this concept of reliability.

Therefore, two values need to be considered [1]:

• limit function G: G = R− S (R: resistance and S: solicitation)

• reliability index β, from which the failure probability p f can be directlydetermined

Analytically, the failure probability of a structure is defined as follows [1]:

p f =∫ +∞

−∞fS(x) · FR(x)dx (5)

with FR: cumulative distribution function

FR(x) = P(R < x) =∫ x

−∞fR(x)dx (6)

and fS: probability density function

P[a < S < b] =∫ b

afS(x)dx (7)

p f relates to the probability of failure of a structural element subjected to asolicitation S and having a resistance R. S and R can follow different probabilitylaws depending on the nature of the the parameter’s behavior. In the context ofthis project, the behavior of S and R will be limited to follow Uniform, Normal,Poisson or Weibull law. These laws can be expressed analytically, and will bedescribed in section 2.1.

Knowing that G = R− S, the failure probability can also be expressed as

p f =∫ 0

−∞f (G)dG = Φ(−β) = 1−Φ(β) with Φ: normal law (8)

This relation shows that probability of failure p f and the reliability index β aredirectly related and dependent of each other.

7


1.3.2 Structural safety verification

According to the probabilistic approach, structural safety is insured when thefollowing condition is satisfied:

β > βlimit or p f < plimit (9)

This shows that the reliability expressed through β or the failure probabilityp f of an element is compared to a limit value. This limit value fixes a minimalreliability expected by society and shows an expectation of society with respect tothe structures. [1]

In the SIA Standards for new constructions, β is fixed at 4.7, corresponding toa failure probability of 10−6. Guidelines are given for target reliability factors forexisting structures.

The design value defined through a probabilistic approach is defined as:

• solicitationS∗ = Sm · (1 + β · αS · νS) (10)

Sm: mean value, α = σSσG

: influence value and µS = σSSm

: variation coefficient

• resistanceR∗ = Rm · (1− β · αR · νR) (11)

Rm: mean value, α = σRσG

=influence value and µR = σRRm

=variation coefficient

1.3.3 Methodology for determining safety factors in the SIA Standard: interrelationbetween probabilistic and deterministic design

In order to define safety factors prescribed in the Standard, following consid-erations have been done: one equalizes the design values obtained by the twoapproaches, equation(3) = equation(10) and equation(4) = equation(11) obtaing

• solicitationSd = S∗

Sd =SkγS

S∗ = Sm · (1 + β · αS · νS)

⇒ SkγS

= Sm · (1 + β · αS · νS)

8


⇒ γS =Sm

Sk· (1 + β · αS · νS) (12)

• resistanceRd = R∗

Rd =RkγM

R∗ = Rm · (1− β · αR · νR)

⇒ γM =Rm

Rk· (1− β · αR · νR) (13)

It goes without saying that the choice of the parameters β, α, and ν is a key choicefor the determination of the safety factors. A safety factor is a function of theimportance of the variable (defined with limit value p f and the reliability indexβlimit.[1]. The design values determined with the probabilistic approach are sensi-tive to the choice of these parameters. This was observed later in the project whendefining the values for the dam design verification.

All the safety factors in the standard should be determined following the fore-going procedure, even if, according to Professor Bruwiler, ”Les normes SIA sedisctuent autout d’une tasse de cafe-Prof. Bruwiler, Septembre 2014”.

1.3.4 Summary and conclusion

The general expressions for the verification of structural safety are:

• deterministic approach: Sd < Rd

• probabilistic approach: S∗ < R∗

For basic cases, it is efficient and precise to compute probability of failure viathe analytical method. However, for more complex examples with numerousparameters, each one following a different probabilistic law, it gets impossibleto find analytical expressions as well as computing the integrals with equations(5),(6) and (7). To face this shortcoming of analytical tools, a method called MonteCarlo has been developed and widely used, being a reliable method to estimatefailure probability. The Monte Carlo approach and it usefulness in the context ofprobabilistic design in civil engineering structures will be explained in section 2.1.

9


The principles of probabilistic and deterministic design explained in this sectionare summarized in figure 1.

Figure 1.: Summary of probabilistic and deterministic design

10

2

M AT H E M AT I C A L A N D N U M E R I C A L T O O L S R E Q U I R E D F O RA P R O B A B I L I S T I C A P P R O A C H

In this chapter, we will describe the probabilistic approach for designing struc-tures. First, the description of the random variables via probability density func-tion will be discussed. Later, the Monte Carlo method will be explained, as wellas its advantages and limitations.

2.1 define probability density functions and the monte carlo

method

In a probabilistic approach we model the uncertainty linked to the input param-eter such as material properties, applied loads and geometric properties withvarious probability distributions.

For each parameter of interest, its uncertainties are taken into account by as-signing a probabilistic law to that parameter. Depending of the nature of theparameter, a different probabilistic law is chosen. The most relevant ProbabilityDensity Functions (PDF) in the context of civil engineering are exposed in thenext paragraphs.

2.1.1 Uniform Distribution

The probabilitiy function of a uniform distribution is defined as

PDF =1

b− afor x ε[a, b] (14)

A parameter following this uniform law is equally likely when lying in therange between a and b. This type of parameter is not relevant in the context of aprobabilistic design since the value will not vary. [5]

11

2.1 define probability density functions and the monte carlo method

2.1.2 Normal Distribution

PDF = f (x) =1

σ√

2πe−

(x−µ)2

2σ2 (15)

with µ: mean value and σ: standard deviation

The normal distribution is probably the most important one in science andengineering. The central limit theorem states that averages of random variablesindependently drawn from independent distributions are normally distributed,which makes the normal distribution attractive to engineers and scientists. Incivil engineering, when providing a construction material, manufacturers givethe mean value and standard deviation characterizing the material strength orsize. These references from the factory can be used by engineers in order to esti-mate the variability of the material resistance and extrapolating it to the structuralreliability.[5]

2.1.3 Weibull Distribution

PDF = f (x) =kλ

( xλ

)k−1· e−(

xλ)

k

(16)

with k: shape parameter and λ the scale parameter of the distribution.

From figure 2, it can be noticed that the shape parameter k has a strong influ-ence on the type of the curve. When k=1, the function looks like an exponentialfunction, and when k is equal to 5, the Weibull is close to a normal law.

Figure 2 shows the Weibull law and the influence of the different parameters λ

and k.

12

2.1 define probability density functions and the monte carlo method

Figure 2.: Probability density function for Weibull distribution-from [5]

The Weibull law is typically used in material science in the domain of failureanalysis of a material over their life span. The Weibull distribution is useful todescribe the ultimate stress of a material and gives an idea about the dispersionof the default sizes present in the material. The shape parameter k is commonlyreferred to as the Weibull modulus. The more similar the defaults of the studiedmaterial, the larger the statistical variance k. Ceramic materials with a homoge-neous distribution of defaults in the material have a high shape parameter k. Inthe context of failure mechanics and defect analysis, the scale parameter λ repre-sents the mean ultimate stress of the sample volume.

The cumulative Weibull distribution is also employed in particle-based methodsto describe the size of particles generated by grinding and crushing operations.In fact, the Weibull distribution predicts a more accurate distribution of smallparticle sizes than the normal distribution. The cumulative distribution functionF(x, k, λ) is the mass fraction of particles with diameter smaller than x, with λ

being the mean particle size and k a measure of the spread of the particle sizes.This particle size distribution and mass fraction estimation with the cumulativeWeibull distribution is applied in the mining industry and in other industrialapplications in order to design the containers and the conveyor belts carryingcrushed material. [5]

13

2.2 the monte carlo method

2.1.4 Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses theprobability of a given number of events occurring at a fixed time interval of timeif these events occur with a known average rate and independently in time. Theprobability mass function, giving the probability of a discrete random variable, isgiven by the expression

PMF =λk

k!· eλ (17)

with λ: average rate and k: number of events occurring in a fixed interval of time

Figure 3 shows the shape of the probability mass function of the Poisson distri-bution for different parameters of the average rate λ.

Figure 3.: Probability mass function for Poisson distribution-from [5]

For sufficiently large values of λ (say λ > 10), the normal distribution and thePoisson distribution converge. Assuming that λ is bigger than 10 in the context ofthis project, we may relate the Poisson distribution to a normal distribution andwill not use it for future application.


This method differs from the probabilistic distributions explained above in a waythat it does not proceed analytically, but is based on repeated random samplingto obtain numerical results. It is widely used in numerical simulations when it is

14


difficult or impossible to use analytical expressions. The methodology of a MonteCarlo simulation is described in the following paragraph.

• Creating the Sample: Random Number Generator

When creating a sample of interest, a domain of definition of possible inputsis established and the inputs are randomly generated from a well-definedprobability distribution.

A few considerations have to be made concerning the random generationof inputs. A computer can only generate uniform pseudo-random numbers.Even if the generated numbers are independently drawn in the sample, thereis a periodicity linked with the random generation, thus generating pseudo-random numbers. However in Python, when using the numpy.random.RandomState()function, the period of return is 219937. The number of Monte Carlo simula-tions in our applications being significantly lower than the period of returnof the sample drawings, we can assume that the numbers are indeed gener-ated randomly.

• Running the Numerical Model

A deterministic condition is set in the loop.In the context of this project,the maximum strength in a structural element (resulting from randomlydistributed variables) is compared to the resistance of the material. If thestrength exceeds the resistance, a counter implemented in the code increasesby one. The model is run as many times as Monte Carlo drawings want tobe done, starting with the number of drawings of N = 10 to N = 1000000 ifthe computer power is sufficient.

• Number of simulations N required

Lemaire gives an estimation of number of simulations N that have to be donein order for the results to converge. For a 95 % reliability of the probabilityof failure value p f = 10−n ⇒ N = 10n+2,3

this means that for a failure probability of

10−6 ⇒ N = 108,9 samples are required in order to achieve good accuracy of results

• Analysing the Data

15

2.3 advantages and limitations of using monte carlo

Through the counter implemented in the code, the number of failure casesin known at the end of the simulation. The probability of failure is thenobtained by dividing by the total number of Monte Carlo simulations N.

2.3 advantages and limitations of using monte carlo

The main advantage of Monte Carlo is its easy numerical implementation, espe-cially for complex cases where the analytical expressions are too complicated. Theresults are reliable and accurate.However, it may take a lot of time depending on the complexity of the problemand the number of samples drawn.

16

Part II.

The application of probabilistictheory: from a basic example in

Python to a real structure in Akantu

17

3

A N I L L U S T R AT I V E E X A M P L E O F T H E M O N T E C A R L OT H E O RY I N P Y T H O N

The goals of this example were to

• learn how to program in Python

• estimate the failure probability of a steel bar subjected to an axial force usingthe principles of Monte Carlo theory

This basic example is relevant because it allowed us to study how the accuracyof Monte Carlo simulations evolves with the sample size and how the resultingfailure probability converged to a value after sufficient simulations were done.

3.1 evolution of the accuracy of the results

3.1.1 Parameter description and example considerations

First, the parameters of the beam are

• the applied force F

• the cross-sectional area of the bar A

• the tensile resistance of the steel bar R

Different cases were considered: each parameter was considered following ei-ther Weibull or Gaussian distribution, which resulted in eight different possiblecombinations (each one of the three parameters can follow two different laws).This allowed to compare the different outputs and influence of the laws on theresults.

Only one single case out of the eight studied cases will described in this report.

18


The mean and standard deviation chosen for the inputs to define the state ofthe beam are

• for the applied force: ν = 200′000 N and σ = 2000 N

• for the cross-sectional area of the bar A: ν = 700 mm2 and σ = 0.2 mm2

• for the tensile resistance of the steel bar R: ν = 330 MPa and σ = 10 MPa

The force F is chosen to follow a Weibull law with a shape parameter λ = 6 andthe area A and the resistance Rare chosen to follow a Gaussian law. The stress σ

in the steel bar subjected to tension is given by equation 18.

σ = F/A (18)

Comment: The choice of the force following a Weibull distribution has onlybeen done for illustrative purposes. It does not make physical sense for a force tofollow a Weibull distribution. However, to stay as close as possible to the reality,the shape parameter k of the Weibull distribution is chosen in order to have aprobability density function as close as possible to a normal law.

3.1.2 Methodology

In order to implement the Monte Carlo theory, a loop is introduced in the code inorder to compare for each loop passage the tensile strength σ induced in the barwith the tensile resistance R of the bar. This methodology is deterministic sinceit compares two values and draws meaningful consequences in terms of failureprobability and reliability of the bar.

3.1.3 Simulation results and outputs given by the Python code

The results obtained on the Python interface for this example are given in thissection. Since the goal of this example was to study the evolution of accuracy ofthe results with the sample size, different results related to different Monte Carlosimulations are shown in the figures below, as well as the associated failure prob-ability.

Starting with one simulation (N=1), the number of simulations was progres-sively increased to N=10e7, being the limit of the Virtual Box installed on ourcomputer. The outputs are shown in the figures 4,5,6,and 7.

19


(a) N=1, p f = 0 (b) N=10e1, p f = 0

Figure 4.: Outputs for N=1 and N=10

(a) N=10e2, p f = 0 (b) N=10e3, p f = 0

Figure 5.: Outputs for N=10e2 and N=10e3

20


(a) N=10e4, p f = 0 (b) N=10e5, p f = 2e− 5

Figure 6.: Outputs for N=1024 and N=10e5

(a) N=10e6, p f = 1.1e− 5 (b) N=10e5, p f = 9.1e− 6


The different figures shown here above give a good idea about the evolutionof the accuracy of the failure probability estimation with respect to the numberof Monte Carlo simulations (resulting in the sample size). At the very beginning,when the number of simulations is still low, the bar chart illustrates well thestudied case: in fact, it can be seen visually that the bars from the resistanceand stress chart do not intersect in the first four images. With the number ofsimulations increasing, the chart starts taking the appearance of a continuousnormal law and the areas of the two curves do intersect, giving a first estimationof the failure probability.

21


3.1.4 Convergence rate analysis

The results shown in the figures from the previous section are summarized in thetable 1.

Table 1.: Summary table

Number of simulations N Failure probability p f

1 010e01 010e02 010e03 010e04 010e05 2.0e− 0510e06 1.1e− 0510e07 9.1e− 06

Theses results can be visualized in figure 8.

Figure 8.: Convergence rate analysis

This plot renders the evolution of the Python outputs with the increase of thenumber of Monte Carlo simulations and brings to light the convergence of the

22

3.2 study of the accuracy of the monte carlo method

output values, converging to a probability of failure of p f = 9.1e− 06.

The failure probability of the steel bar has been hereby computed with a codeimplemented in Python and plausible results have been found through a conver-gence analysis. It is now necessary to study the accuracy of the results predictedby the Monte Carlo method compared to the analytical results. Therefore, another example has been done and will be explained in the following section.


The goal of this example was to estimate the failure probability of a basic examplevia three methods, being the

• analytical method

• approximate method proposed first when typing failure probability on Googleand widely used for most civil engineering practises

• Monte Carlo method

The results obtained with the three methods will be compared and relevantconclusion can be drawn concerning the accuracy of each of the approaches.

The same model as in section 3.1 was chosen: a steel bar subjected to axial ten-sion. The stresses induced in the bar are governed by equation 18.

For this example, the stresses in the bar are following a normal distributionwith a mean value of µ = 286MPa and a standard deviation of σ = 14.29MPa.The resistance is following a Weibull distribution with the scale parameter chosento be λ = 360MPa and a shape parameter k = 20.

3.2.1 The analytical method

The exact solution of the failure probability is analytically defined as

p f =∫ ∞

−∞fs(x)FR(x)dx (19)

23


Figure 9.: Failure probability p f computed analytically

Since the computation of the integral involves a Normal and a Weibull distri-bution, it cannot be done analytically. Thus, it will be done numerically using thePython function scipy.integrate.quad.

We found a failure probability of

p f = 1.550 · 10−2

3.2.2 Approximate method

In order to confirm this result, it was of interest to use a different method toapproximate failure probability. Found in the course [4] and in numerous otherreferences in literature, the expression for an approximation of computing failureprobability is given in equation 20. This integral corresponds to the intersectionarea of the two probability density functions. The two PDF are shown in figure10, the green line representing the stress and the blue line the resistance. The redline delimits the area of the intersection where the integral will be computed.

p f =∫ x

−∞fR(x)dx +

∫ ∞

xfs(x)dx (20)

x: the value where the probability density of both resistance and stress functionis the same:

fs(x) = fR(x) (21)

24


Figure 10.: Approximate calculation of the failure probability p f

By using the same numerical integration tool in Python as for the previoussection, we found an estimation of the failure probability of

p f = 8.788 · 10−2

3.2.3 The Monte Carlo method

Similar to the example in the previous section, for each Monte Carlo drawing, thestresses σ in the bar are compared to the resistance. A counter is set in order tocount the number of times the stress in the bar exceeds the resistance.

Figure 11 shows the distribution for the stresses and the resistance.The sample size for this Monte Carlo simulation was fixed at N = 105 in order

to limit the computation time. The failure probability was calculated by dividingthe number of failed specimens by the total sample size N. For this example, wefound a failure probability of

p f = 1.502 · 10−2

25


Figure 11.: Failure probability calculated with 105 Monte Carlo simulations

3.2.4 Comparison between the three different methods

To estimate a failure probability of the order of 10−2, according to Lemaire’s for-mula, the sample size N is suitable. On the other hand, the approximate solutiongives a result of an order of magnitude greater than the other results. This showsthat the approximate method is very conservative.

It is of interest to compare the different ways to calculate the failure probabilitywhen the target failure probability decreases. This was done by increasing thescale parameter λ, all the others parameters remaining the same. This causes ashift to the right of the resistance curve.

The results are summarized in the table 2 with a fixed number of Monte Carlosimulations N.

Table 2.: Influence of the λR parameter on failure probability

µs σs λR kR p f analytical p f approx. p f MonteCarloMPa MPa MPa - - - -

286 14.29 360 20 1.55e− 02 8.79e− 02 1.47e− 02286 14.29 400 20 1.92e− 03 1.86e− 02 2.07e− 03286 14.29 460 20 1.17e− 04 2.05e− 03 8.00e− 05286 14.29 500 20 2.21e− 05 5.24e− 04 5.00e− 05286 14.29 560 20 2.29e− 06 7.91e− 05 0.00e + 00

This example shows accurately how the Monte Carlo simulation can underesti-mate the failure probability if the target probability is very small. In that case, in

26


order to increase accuracy of the solution, a larger sample size is necessary. Thefact that for λ = 560MPa none of the samples failed shows that the sample size isnot big enough.

Another approach would be to change the sampling procedure by focusing onthe zone of interest e. g. the failure zone (in the context of this example). Thisconcept is called importance sampling or instrumental density and is explainedin a more explicit manner in the course [8].

The basic example of a steel bar in tension gave us the possibility to learn howto

• code in Python

• implement a loop for the Monte Carlo simulation

• compute failure probability analytically using mathematical functions de-fined in the theoretical part of this report

• estimate failure probability with the Monte Carlo methodology and imple-ment a code for the Monte Carlo simulations

• compare Monte Carlo results with the analytical solutions and draw conclu-sions

• evaluate the accuracy of Monte Carlo results with respect to the sample size

We also realized that the failure probability approach proposed in numerousreferences and most civil engineering references has to be treated with cautionsince it is only accurate for small probabilities. In fact, the idea that the failureprobability is equal to the area of intersection under the S and R curve is onlyright for p f < 10−10 and should not be systematically used. Instead, the correctmethod proposed by equation 19 should be kept in mind for a general case.

The next step is to use Akantu via the Python interface in order to estimatefailure probability of more complex cases using the basic principles explained inthe above section.

27

4

A N I L L U S T R AT I V E E X A M P L E O F T H E M O N T E C A R L OT H E O RY U S I N G A K A N T U

As a continuation of the previous example described in chapter 3, the idea isto add difficulty and estimate the failure probability of a beam element (as op-posed to a bar element in previous example). To face the challenges linked withstructural mechanics, the open source software Akantu is used through a Pythoninterface. A simple example was chosen in order to be able to check the resultsanalytically with the equations for basic beams.

The goal of this examples is to

• learn how to use Akantu from the Python interface

• estimate failure probability of a beam with the Monte Carlo method

4.1 getting started in akantu

In order to learn the Akantu’s operation and the functions important for the restof the project, we did a couple of basic examples to get familiar with Akantu. First,we made examples for a static and dynamic response of a beam, then did someimplicit and explicit simulations. We also learned the main differences betweenthe solid and structural mechanics part and the relevant functions that we woulduse for the future example. We visualized most of our result in Paraview [13].

4.2 the model

The case of a simply supported beam is considered. The beam is composed oftwo elements (3 nodes) and a force is applied at the middle of the beam (on themid-node). The beam has a length of 10m.

The varying parameters were chosen to be:

28

4.3 the analytical method

• the applied force F, following a normal with ν = 1000 N and σ = 10 N.

• Young’s Modulus E, following a normal law with ν = 2e10 MPa and σ = 2e7MPa.

• Moment of intertia I

• the resistance of the beam R, following a normal law with ν = 2430 MPaand σ = 2.4 MPa.

4.3 the analytical method

The moments resulting in the steel beam can be easily computed using first princi-ples of statics. This was useful in order to check the results obtained with Akantu.

4.4 the monte carlo simulation

Similarly to the previous example, a loop is implemented with a comparison ofthe stress in the bar with the beam resistance. Each time the solicitation in thebar exceeds the resistance, the counter increases by one. That way, the probabilityof failure can be computed by dividing the failed cases with the total number ofMonte Carlo drawings.

We started with 10 simulations, increasing to N = 10e6 simulations, being thelimit of a reasonable computation time on the Virtual Box.

4.5 results

The results of the simulations are shown in the figures 12,13,14. The evolution ofthe sample distribution can be clearly seen on these figures.

29

4.5 results

(a) N=10e1, p f = 0 (b) N=10e2, p f = 0


(a) N=10e3, p f = 9.0e− 3 (b) N=10e4, p f = 7.2e− 4


(a) N=10e5, p f = 9.0e− 4 (b) N=10e6, p f = 9.89e− 4


30

4.5 results

The convergence rate analysis of this example is shown in the figure 15. Theconvergence is not as smooth as the for the example of the steel bar in tension,but it nevertheless converges to a value of 9.89e− 4.

Figure 15.: Convergence rate analysis

The failure probability of this beam for the studies load case is estimated to bep f = 9.89e− 4.

This example was a good introduction to Akantu and a key element for theimplementation of case of the dam in the next chapter.

31

5

U S I N G A K A N T U F O R A N A P P L I C AT I O N O F T H E M O N T EC A R L O T H E O RY

5.1 introduction

Even if non-sampling-based methods such as polynomial chaos are more precisethan the sample-based Monte Carlo method, the Monte Carlo method presentsa clear advantage when it comes to complex geometries and non-linear systems.Since the overarching goal of this project is to estimate the failure probability ofa dam, the Monte Carlo is the most accurate method being able to face the chal-lenges related to the complexities of the future problem.

Having used Akantu via the Python interface for a basic example to estimatethe failure probability of a structure, we are now eager to apply to knowledgeacquired so far in order to study a complex structure. In order to do that, thestructure of a dam has been chosen for the purpose of this semester project. Theidea is to create the mesh of a dam embedded in rock, define key parameterswhich follow a given probabilistic law and finally apply the Monte Carlo theoryin order to compute the failure probability of a structure.

Through a loop over the number of Monte Carlo drawings, each drawing com-pares the maximum displacement computed via Akantu with the allowable dis-placement, the maximum tensile strength in the concrete with the resistance intension, the maximum compressive strength with the resistance of concrete incompression.For simplification purposes, it has been assumed that the dam fails when thestress at least one element exceeds the resistance of the concrete. We are awareof the fact that this is not totally corresponding to the reality since a local failureof concrete would not necessary imply a total failure of the dam. However, thissimplification allows a quick assessment of the reliability of the dam and gives agood approximation of the failure probability of the structure.

32

5.2 considerations about the model


The mesh of the dam set up for the simuation is taken form another project of anarc dam in Canton Bern, Switzerland. The dam does not exist yet and the geomet-rical properties have been taken from another semester project at the Laboratoirede Constructions Hydrauliques (LCH) at EPFL.

The dam is modelled as a monolithic section, without joints. The connection ofthe dam with the foundation and the rock is assumed to be rigid.

5.2.1 The methodology and principles implemented in the code

The analysis of the dam was done using Akantu through the Python interface.Akantu was used for static solid mechanics analysis.

For each Monte Carlo simulation, the principal stresses in each element of thedam are computed via a function directly computing the Eigenvectors and Eigen-values. Then, the maximum of the principal stresses of all the elements is com-pared to the tensile resistance of concrete in order to determine whether or notfailure occurs. The same procedure is implemented for the compressive strength.A counter is set up in the code and increases each time the tensile strength inone element exceeds the concrete resistance or the compressive strength in oneelement exceeds the concrete resistance in compression. Finally, the total numberof failure cases is divided by the number of Monte Carlo drawings in order todetermine the probability of failure.

The displacement of the midpoint of the dam is computed for each MonteCarlo drawing and compared to an maximum allowable displacement, an arbi-trary value fixed by the engineer. Even if the displacement is not relevant interms of structural safety (tensile stresses and compressive stresses will deter-mine whether the structure will fail or not), it is interesting to study the evolutionof the displacement over the Monte Carlo drawings.

5.2.2 The mesh setting up

Our aim was to model a real example, with a complex 3D geometry due to adouble curvature of the arch and the presence of the cantilever, as well as theasymmetry of the valley flanks.

33


This complex geometry was approximated with 8 2D arcs of constant depthand of a circular shape. In reality, elliptical arcs or logarithmic spiral arcs canbe chosen to optimize the incident angle with the foundation. To model theconnection between the dam and the bedrock, a raw geometry of the valley hasbeen added to the existing model. The mesh was done with the open sourcesoftware GMSH.[14]

The 3D mesh has a tetrahedron geometry. First and second order computationshave been done (4 and 10 quadrature points respectively).

The aspect of the mesh in GMSH can be seen in figure 16.

Figure 16.: Aspect of the mesh in GMSH

We set two different mesh sizes: a fine one for the dam and a coarse one for thebedrock. In order to define the optimal mesh size, a sensitivity analysis has beendone: by decreasing the mesh size, we analysed the evolution of the maximaldisplacement and of the computation time required for each simulation.

First, this had to been done for the dam alone in order to find the optimal meshsize of the dam. The results of the sensitivity analysis are shown in figure 17 and

34


18.

The loads applied on the structure are the dead load of the dam and the hydro-static pressure. It is interesting to see that the relation between computation timeand the number of nodes on the mesh is almost linear.

Figure 17.: Influence of dam mesh size in m (on the x axis) on maximal displace-ment in m (on the y axis)

Figure 18.: Influence of dam mesh size in m (on x axis) on computation time in s(on y axis)

Then, fixing the mesh size of the dam, we varied the mesh size of the bedrockand performed the same calculations. The final mesh size has been chosen by do-ing a compromise between execution time and precision.The optimal mesh sizesadopted for the Monte Carlo simulation are: for the dam hdam = 4m and for the

35


bedrock hrock = 40m.

Figure 19.: Influence of bedrock mesh size in m (on the x axis) on maximal dis-placement in m (on the y axis)

Figure 20.: Influence of bedrock mesh size in m (on the x axis) on computationtime in s (on the y axis) and the number of nodes

It is important to note that this analysis has been done by considering only thedisplacement. For a real life application it would be essential to perform the sameanalysis considering the evolution of the stresses as well. In fact, the thickness ofthe dam is in the range of 10m, with a mesh of 4m by looking at the cross-sectionwe realize that the mesh is rather coarse and the stresses caused by the bendingmoment in the cantilever may not be accurate cf. figure 21.

Another issue occurred at the contact between the dam and the foundation. Inour simplified geometry there is a sharp edge at this interface dam-foundation,

36


which causes a local stress concentration cause by the model, but not necessarilyoccurring in reality. Therefore, our model is accurate to compute displacements,but not to compute stresses in the vicinity of the foundation and on the interfaceon the side between the dam and the bedrock, as it can be seen in figure 21.

In order to avoid these stress concentrations at the interfaces between bedrockand concrete dam, an additional physical volume has been created, which in-cludes only the central part of the dam. In the Monte Carlo loop, the structuralanalysis and stress comparison was only done in this inner region.

Considerations for future work: For a better estimation of the stresses over theentire domain and for a more accurate model, the mesh model should be revisitedand sharp edges should be avoided, and a finer mesh should be chosen.

Figure 21.: Stress Concentration at the interface dam - foundation

5.2.3 The materials: rock and concrete

The materials are assumed to behave in a linearly elastic way and are describedin the model by their Young Modulus E, Poisson Ratio ν and Density ρ.The nu-merical values are defined in chapter 5.3.

37


The density of the rock is set to 0 since we are only interested in the stressesand displacements in the concrete and we are not performing a dynamic analysiswhere the rock mass have an influence. Furthermore, we do not want to takeinto account the deformation due to the wight of the rock-mass and can thereforeneglect the dead load of the rock for the purposes of our project.

5.2.4 The loads

The loads considered are the dead load of the dam and the hydrostatic pressuredue to the water considering a full lake (worst case scenario in term of magnitudeof hydrostatic pressure).

The dead load of the dam has been applied through a force vector at thecentre of every element. The hydrostatic pressure is applied on the dam atthe upstream face and is applied perpendicular to the surface via the functionmodel.applyHydrostaticPressure defined in Akantu and used through thePython interface.

38

5.3 the monte carlo simulations

5.3 the monte carlo simulations

5.3.1 Risk scenarios

The considered risk scenario is the lake entirely filled with water. We did notconsider the scenario of the empty lake. We did not consider any pressure in-duced by the depositions of sediments or by the formation of ice. The effect ofthe temperature was not taken into account.

The height of the water is fixed at 1720m (the altitude of the crown of the dam)and will not vary with the different simulations, since it is upper boundary forthe water height and the worst possible case for the considered scenario.

5.3.2 Failure criteria: deterministic values

A first criterion is necessary for the maximal allowable tension in the concrete.From [9], we fixed this value to a 2MPa resistance. For each Monte Carlo simula-tion, the maximum tension stress in the concrete will be compared to the 2MParesistance, and each exceeded value will be kept as a failure case.

The second criterion is necessary for the maximal compression in the concrete.Again, from [9], we defined this value as 8MPa.

These two criteria are the key criteria for structural resistance and will be usedin order to evaluate structural reliability.

Since we are not dealing with an extreme scenario (ultimate failure scenario),we chose to impose a more conservative characteristic resistance than the one sug-gested by Prof. Schleiss in [9]. This is also related to the fact that, for the givenrisk scenario, we want no damage at all to occur and assume that an exceededresistance value in one single element corresponds to the failure of the structure(conservative approach). The tensile resistance of concrete was fixed to be around0.3MPa (one failure occurred for this tensile resistance). The compressive strengthis chosen to be 6MPa.

Furthermore, as part of this semester project, we intended to show the evolutionof the failure probability with the evolution of the sample size. Due to the limitedcapacity of the Virtual Box and thus the limited number of possible simulations, itwas necessary to lower the resistance (compared to the resistances defined by [9]in order to assure at least on failure case (otherwise no failure at all would have

39

5.4 results

Table 3.: Parameter describing probabilistic distribution

item unit µ σ

Edam GPa 20 1ρdam kg/m3 2500 100Erock GPa 20 2

occurred, hence no failure probability could have been computed, which wouldbe at no interest in the context of this project).

5.3.3 Material parameters used and description of the nature of these parameters

Two materials have been defined in the model: concrete and bedrock.

In this project, only the parameters related to the concrete of the dam are as-sessed with a probabilistic approach. We chose Young’s Modulus E and density ρ

of the dam following a normal law, which accurately corresponds to reality sincethe concrete quality in the dam is of high variability.

5.3.4 Sample size

The sample size required to estimate a failure probability in the order of magni-tude of 10−6 (which corresponds to the estimated failure probability for an im-portant structure and equivalent to a β = 4.7 according to SIA Standard) wouldrequire sample size N = 108. Because of the limited computation time it was notpossible to produce such a big sample. The number of computations achievedwas N = 340000.

5.4 results

In the following paragraph the results of the Monte Carlo simulation will bediscussed. First the distribution of the maximal displacements and the stresseswill be presented, later we will study the evolution of the failure probability withincreasing sample size.

40

5.4 results

5.4.1 Distributions

Figure 22 shows the distribution of the maximal displacement in the central partof the dam. The average displacement is d = 1.07 mm and its standard deviationis 0.104mm.The average value is just above the arbitrary limit we imposed to bedmax = 1mm. N.B.:This arbitrary value has been chosen in order to get an interestingexample and does correspond to displacement boundary values in reality

Figure 22.: Distribution of the maximal displacement

The distribution of the maximal compression is shown in figure 23. It shows ashape similar to the distribution of the maximal displacements.The average max-imal compressive stress is σc = −2.95 MPa and it is above the resistance in com-pression of the concrete σr = 7 MPa, the standard deviation is 0.0107 MPa.

41

5.4 results

Figure 23.: Distribution of the maximal compression

Figure 24 shows the distribution of the maximal tensile stress present in thestructure. The mean value is σt = 0.29 MPa with a standard deviation of 0.00025MPa. This value has a great importance because the tensile strength of concreteis very low and is in most cases the weakest link in a failure scenario. Since thereis no reinforcement in an arch dam, the structure is supposed to be solely undercompressive stress and tensile stresses should be absolutely avoided. The geome-try of this dam should be optimized in order to have compression everywhere inthe structure and no significant tensile stress should appear in the concrete.

An interesting feature is the very sharp lower boundary in the distribution. Thisbehaviour is unexpected and physically unrealistic. Further study should be doneto asses the origin of this phenomena, whether is coherent with the real behaviourof the structure or it is caused by a computational error or by a singularity in themesh.

42

5.4 results

Figure 24.: Distribution of the maximal tension

5.4.2 Interdependences

It is relevant to study which input parameter mostly affects the behaviour of thestructure. In a qualitative way we can say that the smaller the E moduli of bothdam and bedrock, the greater the maximal displacement. On the other hand theinfluence on the stresses is difficult to predict, we could imagine that a lower Emodulus of the bedrock would cause lower stresses in the structures, due to thehyperstaticity. This hypothesis will be verified in the following paragraph.

Figure 25 clearly shows how the maximal displacement directly depends onthe E modulus of the rock, while for the other parameter no clear dependency isnoticed and the points are scattered around their average value.

43

5.4 results

(a) Maximal displacement vs Edam

(b) Maximal displacement vs Erock

(c) Maximal displacement vs ρ

Figure 25.: Parameters influencing the maximal displacement

44

5.4 results

Figure 26 shows the distribution of the maximal tensile stress as a function ofthe input parameters. The tensile stress is reduced when the E modulus of therock is lower. It can be seem that there is a small increase in tensile stress whenthe E modulus of the dam decreases. No clear interdependence can be seen withrespect to the density of the concrete.

Figure 27 shows the plots of the maximal compressive stress as a function ofthe different input parameters. Again we see that the E modulus of the rock hasthe greater influence on the values of the stress compared to the other parameters.This time, a clear increase in stresses appears when the stiffness of the foundationin decreased. The opposite trend is seen for the E modulus of the dam but witha less clear pattern. Again no clear relation between density and stresses can bededuced from this analysis.

For all the considered parameters, we notice a strong relation with the Youngmodulus of the bedrock. This is reasonable, because an arch dam is a hyperstaticstructure and therefore the conditions at the connection with the foundation isof fundamental importance for the behaviour of the structure and thus for theassessment of its safety.

45

5.4 results

(a) Maximal compressive stress vs Edam

(b) Maximal compressive stress vs Erock

(c) Maximal compressive stress vs ρ

Figure 27.: Parameters influencing the maximal compressive stress

46

5.4 results

(a) Maximal tensile stress vs Edam

(b) Maximal tensile stress vs Erock

(c) Maximal tensile stress vs ρ

Figure 26.: Parameters influencing the maximal tensile stress

47

5.4 results

5.4.3 Convergence of the failure probability

Using the data obtained with the simulations, we were able to asses the failureprobability of our structure, by setting for criteria a tensile strength of σsr = 0.3MPa and a compressive strength of σcr = 7 MPa.

The value of the tensile strength has been chosen for didactic reason in orderto have at lest one failure. Theoretically, we could also have imposed no tensileresistance at all, but then the failure probability would have been p f = 1 (therewould certainly at least one element failing the criterion). Moreover, due to astress redistribution after a crack opening, imposing zero tensile resistance to theconcrete would not be accurate.

Figure 28 shows the evolution of the failure probability with increasing samplesize. The failure probability p f = 2.94 · 10−6 is not accurately estimated, becausejust one sample failed. A larger sample size would be necessary, as discussed inchapter 5.3.4.

Figure 28.: Evolution of failure probability with sample size: Stress criterion

48

5.4 results

Figure 29.: Evolution of failure probability with sample size: Displacement crite-rion

To illustrate the evolution of the failure probability with increasing sample size,we decided to introduce a fictive failure criterion by setting an upper limit to thedisplacement capacity of the structure dmax = 0.001 m. This criterion is fictive inthe sense that displacement will not govern structural safety of dam. Figure 29

shows very clearly that p f slowly converges to its asymptote, and estimates thefailure probability of the dam to be p f = 0.264.

49

C O N C L U S I O N

• Safety coefficients ’Safety coefficient’ does not have an intrinsic meaning, ithas to be associated with the concept of characteristic value to give it anysens. The reliability theory predicts a close relation between these two con-cepts, being the basic of deterministic design but hence having probabilisticbackground. Improvements can result from the reliability theory, for exam-ple by rewarding a better production quality with a decrease in the partialcoefficient associated with the concerned variable [7]. It has also been shownthat it is important to know the concepts behind the definition of the safetycoefficients in order to be able to use them appropriately and adjust them inthe case of existing structures.

To conclude, it is important to keep in mind that a safety coefficient onlydenotes a number, which, associated with data selection, a failure scenarioand a rule, generally results in a satisfactory design. But this coefficient doesnot measure safety, and all the ignorance and uncertainties of the engineersare masked behind this coefficient. Safety can only be evaluated with a goodknowledge of these coefficients and an sensitivity analysis of the parametersof the model.

The most universally adopted method for quantifying reliability dependson the computation of probabilities. [7]. Even if the probabilistic methodallows to accurately quantify the failure probability of a structure and givean estimation about how close the structure is to failure, this probability p fis still a number. Theoretically, for a structure that has a probability of failureof 10−6, it theoretically means that if one would build 1’000’000 houses withthe same materials and same workers, one of the houses should fail due toa particular event.This absurd example shows that there is an undeniable gap between thetheory of reliability in buildings (as powerful as it will ever be) and the realstate of a structure that will never be able to be quantified with engineeringtools.

• What we have learnt from this project: the biggest challenge of this projectprobably lies in the use of the numerical tools and the implementation of

50

5.4 results

the code. Learning new concepts in programming in Python and learninghow to use Akantu through the Python interface was a very interesting andchallenging part. Throughout the semester, we also faced the limitations ofcomputers and programming (e.g. divergent results because of inappropri-ately defined interface conditions) and got aware of the engineer’s respon-sibility: design problems require a good understanding on the part of theengineer in charge of the design project. Structural safety and reliability hasto be guaranteed to the building user and these concepts need to be welldefined and understood by the civil engineer. Numerical tools may help tosolve the technical part, but the fundamental understanding lies in the handof the engineer. In this perspective, immersing ourselves in the probabilisticapproach in civil engineering design made us aware of the challenges hid-den behind structural safety and reliability. We discovered a new approachfrom a theoretical point of view by reading through research papers, beforeapplying this knowledge to estimate failure probability of a real structure.What made this project interesting was its progression: starting from a sim-ple steel bar, we were able to finally implement a code and run simulationsfor a dam.

• Considerations for future work in the topic: for more accuracy and a moreefficient computation of probability of failure, the approach proposed bythe course [8] would be a good starting point for future imporvements. Infact, the methodology proposes more specific analysis (more samples in thecontext of Monte Carlo) in the zone of interest. In the context of our damstructure, it would mean increasing the number of samples in the failurezones (middle of the dam) and decreasing the number of sample drawingsin the irrelevant zones.

Suggestions to improve our work: to simplify the problem, we made theassumption that Young’s modulus E of the rock is constant through thebedrock mass. However, this is not strictly correct geotechnically speaking.In reality, E is higher in the depth of the rock than on the outer surface ofthe rock (because of squeezing and confining effects). For more accuracy inour model, this E modulus should be appropriately defined across the mesh.

Another interesting study for future work would be the assessment of thesafety of an existing structure under a rare event, e.g. seismic loading.

51

5.4 results

We want to address a special thank you to our teaching assistant FabianBarras for great help, support and patience granted throughout the

semester.

52

B I B L I O G R A P H Y

[1] Eugen Bruhwiler. Polycopie du cours Securite et Fiabilite. EPFL-ENAC-MCS,Septembre 2012.

[2] Eugen Bruhwiler. Polycopie du cours Structures existantes: Examen et interven-tions, notions de base. EPFL-ENAC-MCS, Septembre 2014.

[3] Eugen Bruhwiler. Polycopie du cours Structures existantes: Examen et interven-tions, Chapitres choisis. EPFL-ENAC-MCS, Fevrier 2015.

[4] Laurent Vulliet. Polycopie du cours Fiabilite et securite des systemes civils. Partie1, EPFL-ENAC-LMS, Juin 1997.

[5] Wikipedia: http://www.wikipedia.org

[6] Maurice Lemaire. Article scientifique Approche probabiliste de dimensionnementet mondelisation de l’incertain et methode de Monte Carlo. Reference BM5003,publie le 10 avril 2014.

[7] Maurice Lemaire and Alaa Chateauneuf. Structural reliability. London ISTE,2009.

[8] B. Sudret, Slides from lecture Structural reliability and risk analysis, Simulationmethods. ETH-Department of civil, environmental and geomatic engineering-Chair of risk, safety and uncertainty quantification, November 2014.

[9] Anton J. Schleiss et Henri Pougatsch, Les barrages. Traite de Genie Civil, vol-ume 17. Presses Polytechniques et Universitaires Romandes, 2011.

[10] Scientific modules for Python, including NumPy and Matplotlib:http://www.scipy.org/

[11] NumPy for MatLab users: http://mathesaurus.sourceforge.net/matlab-numpy.html

[12] MOOC’s for Python: https://www.coursera.org/course/scicomp

[13] http://www.paraview.org

[14] hhtp://geuz.org/gmsh

53

Bibliography

[15] https://www.python.org/

[16] http://lsms.epfl.ch/akantu

54

a probabilistic approach to design civil engineering ......1.3 the probabilistic approach 6 2...

Documents