Reliability-based ultimate limit state design in finite element methods

Transcription

1 Delft University of Technology Faculty of Civil Engineering and Geosciences Section Geo-Engineering, Civil Engineering Reliability-based ultimate limit state design in finite element methods Author: Stefan Kamp Graduation Committee: Prof. ir. R.B.J. Brinkgreve, TU Delft - Section Geo-Engineering Prof. dr. M.A. Hicks, TU Delft - Section Geo-Engineering Ir. K. Reinders, TU Delft - Section Hydraulic Engineering Ir. A. Van Seters, Fugro Geoservices B.V. To be publicly defended on November 29, 2016

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3 Preface In February I started my research at Fugro Geoservices B.V. in Leidschendam. First of all I would like to thank the people at Fugro for this opportunity and the interesting research topic. I would like to thank my colleagues at Fugro who were all friendly and cooperative. During this research I had a lot of help from my daily supervisor ir. van Seters. He helped to keep me on the right track and to improve my research. Thanks to his experience and expert opinions during our meetings I was able to get the most out of my research. I would like to thank the rest of the committee for their involvement. All members of the committee shared their experience and knowledge in order to get the most out of my research. During this process ir. Reinders showed great interest in the topic and helped to improve my research with enthusiasm. I would also like to thank prof. ir. Brinkgreve and prof. dr. Hicks for their involvement and expert opinions during out meetings. Furthermore I would like to thank my family and friends. My family always supported me during my years as a student. Thanks to their support and faith I was able to keep going through the years. Finally I want to thank my friends for these joyful years as a student. During my study many great memories were made and many great times are still ahead. Stefan Kamp November 2016

4 Summary In recent decades finite element methods have become a common tool to design geotechnical structures. Soil effects can now be included, which are not accounted for by traditional methods and design standards. In most European countries geotechnical structures are designed according to Eurocode 7. This design standard described the general principles and requirements and ensures the safety of structures by means of a partial factor based methodology. The partial factors are calibrated on experience and can be applied on a wide variety of geotechnical applications. A problem which is encountered with Eurocode 7 is the difficulty of correctly applying these partial factors. For complex structures, where a combination of failure mechanisms occurs, Eurocode 7 is difficult to adopt. This is due to the fact that this design standard considers geotechnical structures with one single failure mechanisms. Ultimate limit states where both strength and stiffness properties are dominant, are therefore problematic to assess. A complex combination of failure mechanisms is for example encountered for a cantilever retaining wall on foundation piles. Bending moments in the piles occur due to the combination of two effects. First, strength properties behind the wall induce a rotation of the wall, which generates bending moments in the piles. Second contribution is due to soil deformations, which induce a lateral pressure on the piles and cause additional bending moments in the pile. The latter contribution partly depends on the soil stiffness and soil-structure interaction. Due to the combination of these effects, it is complicated to properly assess the reliability of the structure according to Eurocode 7. Recent developments allowed to explicitly define uncertainties with reliability-based methods. By explicitly defining uncertainties, partial factors are no longer required. However, at this point probabilistic methods are usually time-consuming, which is inconvenient in engineering practice. Probabilistic analyses allows to obtain valuable information for the design of geotechnical structures, such as the reliability index. The reliability index or probability of failure provides a more rational and meaningful measurement of the safety than the traditional factor of safety. Eurocode 7 defines different target reliability indices for different Reliability Classes, which are reached by applying the corresponding partial factors. Reliability-based methods allow to derive a specific reliability index for each structure. In order to design a safe structure, the computed reliability index must be larger than the target reliability index in Eurocode 7. In this thesis a straightforward and economic reliability-based method is investigated to design geotechnical ultimate limit states. The reliability-based method used in this research is the Point Estimate Method. First the Point Estimate Method is verified in Plaxis 2D against Monte Carlo simulations by means of three benchmarks. These benchmarks consider typical standard geotechnical structures. Afterwards, the advantages and limitations of the proposed method are shown with a more realistic case study. This case study concerns a cantilever retaining wall on foundation piles. Since this case study considers a complex structure which involves a combination of failure mechanisms, this case study can be ideally used to show the advantages of the Point Estimate Method over Eurocode 7. It is concluded in this research that the Point Estimate Method provides practically similar results as Monte Carlo simulations, but requires significantly less computational effort. Therefore, this efficient and economic reliability-based method is particularly convenient for engineering practice. Furthermore, the Point Estimate Method provides valuable additional information when ultimate limit states are assessed. Instead of a single deterministic Eurocode 7 value, a range of possible results is now obtained in the form of a probability distribution. Moreover, the Point Estimate Method provides results in accordance with Eurocode 7. It is common practice in geotechnical engineering that new design standards are consistent with previous standards. Altogether, the Point Estimate Method is an economic method that allows for a straightforward and efficient derivation of a specific reliability index.

5 Contents Summary List of symbols Abbreviations ii v vi 1 Introduction General Problem definition Research goals Research outline Literature study Design standard Eurocode Serviceability limit state and Ultimate limit state Partial factors Design Approaches Eurocode 7 and Finite Element Methods Uncertainty Sources of uncertainty Coefficient of variation Characteristic soil properties Reliability-based design Reliability analysis Reliability-based methods Point Estimate Method Output uncertainty Comparison reliability-based methods Theory structures Lateral earth pressure Failure mechanisms Important variables Verification Point Estimate Method Introduction Verification methodology Benchmark 1: Slope stability problem Problem geometry and soil properties Normal input distributions Lognormal input distributions Correlated cohesion and friction angle Layered cohesive slope Comparison with Eurocode Conclusion Benchmark 2: Shallow foundation Problem geometry and soil properties Normal input distributions Comparison with Eurocode Conclusion Benchmark 3: Cantilever retaining wall Problem geometry and input parameters Sensitivity Analysis Normal input distributions iii

6 3.5.4 Comparison with Eurocode Conclusion Conclusion Verification procedure Case study: Cantilever retaining wall on piles Problem description Problem geometry and soil properties Sensitivity analysis Point Estimate Method Factor of safety Bending moments Comparison with Eurocode Factor of safety Bending moments Conclusion Case Study Conclusions and recommendations 96 Bibliography 104 Appendices 104 A NEN Table 2b B Random variables C Reliability-based methods D Relevant Plaxis features E Template Point Estimate Method F Benchmark 1: Results slope stability problem G Benchmark 2: Convergence Monte Carlo simulations H Benchmark 3: Convergence Monte Carlo simulations I Benchmark 3: Results PEM cantilever retaining wall J Results Case Study: Sensitivity Analyses K Results Case Study: PEM factor of safety criterion L Results Case Study: PEM bending moment criterion

7 List of symbols α sensitivity coefficient A cross sectional area [mm 2 ] β reliability index [ ] c cohesion [kp a] COV coefficient of variation (COV = σ/µ) E structural or soil stiffness (Young s Modulus) [kp a] EI bending stiffness [knm 2 /m] E[X] expectation of random variable γ volumetric weight [kn/m 3 ] γ unsat volumetric weight of unsaturated soil [kn/m 3 ] γ sat volumetric weight of saturated soil [kn/m 3 ] γ m material partial factor [-] ξ standard deviation unit k n statistical coefficient M bending moment [kn m] or [kn m/m] µ mean value M sf factor of safety [ ] n number of random variables N c, N q, N γ bearing capacity factors ν skewness P i weight for calculation i P f probability of failure V ar(x) variance ϕ angle of internal friction [ ] Φ standard normal cumulative distribution ρ correlation coefficient R resistance S load σ standard deviation σ 2 variance x evaluation points X soil property Z limit state function

8 List of abbreviations BH Brinch-Hansen cdf Cumulative Distribution Function COV Coefficient of Variation (COV = σ/µ) CC Consequence Class DA Design Approach DC Design Combination EC7 Eurocode 7 FEM Finite Element Method FORM First Order Reliability Method HS Hardening Soil MC Monte Carlo MFA Material Factoring Approach LRFA Load and Resistance Factoring Approach LRFD Load Resistance Factoring Design PEM Point Estimate Method pdf Probability Density Function RBD Reliability Based Design RC Reliability Class SLS Serviceability Limit State ULS Ultimate Limit State

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10 Chapter 1 Introduction 1.1 General Geotechnical structures are generally designed according to design standards, such as Eurocode 7. Eurocode 7 describes the general principles and requirements in order to ensure a sufficient level of safety. In this design standard, which is based on a Load and Resistance Factoring Design methodology, partial factors are adopted to account for uncertainties in material properties and loads. With these factors, appropriate design values are obtained for ultimate limit state calculations. The main source of uncertainty in geotechnical engineering is the soil itself due to heterogeneity and insufficient soil data. The partial factors are calibrated on a target reliability index and based upon experience and probabilistic methods. These factors are calibrated such that they can be applied for a wide variety of geotechnical applications. Adequate use of these partial factors ensures that the target reliability index is reached. Over the years much experience is gained with the design of ultimate limit states according to the partial factor method in Eurocode 7. Engineers confidently work with this method for relatively straightforward designs, such as slope stability problems. However, when structures increase in complexity the correct application of partial factors becomes problematic. The safety philosophy as described in Eurocode 7 often assumes constructions where only one failure mechanism is described. Structures in urban areas are often located close to each other and therefore complex failure mechanisms may develop. Due to the interference of several failure mechanisms it may be difficult to distinguish the governing failure mechanism analytically. Correct application of partial factors to achieve a reliable design is complicated or impossible. In order to overcome this problem, traditionally one overall safety factor is used. All uncertainty is captured in one lumped factor which is applied afterwards. This rigorous simplification does not allow to adequately account for different sources of uncertainty individually and it is unclear what safety level is achieved. Nowadays finite element methods have proven to be a useful tool to design geotechnical structures. Originally it was primarily developed to calculate stresses and deformations, i.e. serviceability limit states. More recently, finite element methods are also used for ultimate limit states. An advantageous feature of finite element models is the automatic identification of the governing failure mechanism. This feature makes the use of finite element methods particularly convenient for more complex structures, where it may be impossible to identify the governing failure mechanism prior to the analysis. However, disagreement exists on the use of finite element models in combination with Eurocode 7. As with traditional methods the correct application of partial factors may be problematic. Another development in geotechnical engineering is the use of probabilistic methods in combination with finite element models. In contrast to the partial factor method this approach allows to explicitly define uncertainties in the design. This explicit definition of uncertainties makes the use of partial factors redundant and allows to use the available information more efficiently. With so-called reliability-based methods it is possible to obtain a specific reliability index, which should be higher than the specified Eurocode value for a safe design. However, currently the use of probabilistic methods is not common in engineering practice, as it involves a large effort in man hours. In this thesis an alternative procedure for an extensive probabilistic method is described and analysed. The Point Estimate Method is a relatively straightforward method and provides a good approximation of a full probabilistic analysis and allows to derive the probability of failure. This procedure has been calibrated and used in a complex design situation. 2

11 1.2 Problem definition Difficulties are encountered when complex geotechnical structures are designed for ultimate limit states according to Eurocode 7. These problems are not encountered with serviceability limit states, since partial factors are all equal to unity here. In the case of complex structures with a combination of failure mechanisms there is no clear guidance available in Eurocode 7. Finite element methods in geotechnical engineering allow for a design of more complex structures, since the governing failure surface is automatically identified. Ultimate limit states where only strength properties, such as the cohesion and friction angle, govern the stability are conveniently solved by reducing these parameters in order to obtain an overall factor of safety. However, more complex structures are possibly dependent on both strength and stiffness properties. In these cases a simple shear strength reduction is not sufficient to assess the safety. Figure 1.1: Example of a cantilever retaining wall on foundation piles. Soil-structure interaction significantly affects the behaviour in more complex design situations. This effect is for instance encountered with the design of a cantilever retaining wall on foundation piles. This structure is not only influenced by strength properties, but also by stiffness properties. Bending moments in the foundation piles occur due to contribution of two effects. One part is due to the horizontal pressure against the retaining wall, which is largely a function of the soil weight and the friction angle of the backfill soil. As a result, bending moments occur in the pile due to the rotation of the retaining wall. Furthermore, the soil behind the wall causes settlements and induces a lateral pressure on the piles due to soil deformations. The latter contribution is partly an effect of the soil stiffness and soil-structure interaction. The contribution of both strength and stiffness properties complicates the application of partial factors. A more appropriate way to account for uncertainties in the design is by using reliability-based methods. Reliability analysis allows to explicitly account for uncertainty in parameters with the use of stochastic variables. Input parameters are defined with probability density functions and the use of partial factors is no longer required. Another advantage is that a specific probability of failure and reliability index of a system can be obtained, instead of a pre-determined reliability index which is reached by adopting the partial factor approach. However, other problems are encountered with the use of probabilistic methods. Generally reliability-based methods are not adopted in engineering practice, since they require significantly longer computation times than deterministic calculations. In this thesis a design procedure is proposed where a relatively straightforward probabilistic method is adopted in combination with a finite element program. The possibilities to achieve safe designs without partial factors are investigated. Instead of a safety factor, the safety will be based on the reliability index or probability of failure. This is especially relevant for combined geotechnical structures, since difficulties with the current partial factor method are encountered for these types of structures in particular. Chapter 1.2 MSc Thesis S.P. Kamp 3

12 1.3 Research goals The main research goal of this master thesis is to provide a method for the design of complex geotechnical structures in the ultimate limit state, which combines deterministic analysis and reliability-based design in finite element programs. An important requirement is that this method should be applicable in engineering practice, i.e. straightforward to use and designs can be made within reasonable time. The following research questions result from the main goal: How can little (statistical) information be used to obtain the reliability index in ultimate limit state? How is the output influenced by the number of stochastic variables? Can the Point Estimate Method be used to make geotechnical designs for combined structures? How does the Point Estimate Method compare to other methods (partial factor approach, Monte Carlo)? 1.4 Research outline In Chapter 2 the relevant literature for this research is summarized. The literature study includes the basic principles of Eurocode 7 and guidance on the use of this design standard in combination with finite element methods. Furthermore, several sources of uncertainy in geotechnical engineering, and how to account for these uncertainties, are discussed. General reliability theory and the Point Estimate Method are elaborated in this chapter. Finally, relevant basic theory of geotechnical structures is summarized. Chapter 3 contains the verification procedure of the Point Estimate Method where three benchmarks are analysed. Verification is done by comparing results from the Point Estimate Method with Monte Carlo simulations. Basic principles of the Point Estimate Method are discussed, as well as the verification procedure. Main tools in this verification process are Plaxis, Slide6.0 and Phase2. Subsequently, after verification of each benchmark, the Point Estimate Method is compared with the partial factor method from Eurocode 7. In Chapter 4 a case study is discussed. A combination of geotechnical structures where both strength and stiffness properties affect the behaviour of the structure is assessed. This case study considers a cantilever retaining wall on foundation piles. Shortcomings and advantages of both Eurocode 7 and the Point Estimate Method can be ideally shown with this case study. Chapter 5 summarizes the conclusions that were made in this thesis. Furthermore recommendations for further research are provided. 4 MSc Thesis S.P. Kamp Chapter 1.4

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14 Chapter 2 Literature study This chapter summarizes the relevant literature for this research. Since geotechnical structures are designed according to Eurocode 7, the basic principles of this design standard are highlighted. This involves the application of partial factors and different Design Approaches. Furthermore, the use of Eurocode 7 in combination with finite element methods is discussed. In the design of geotechnical structures one has to cope with many uncertainties, for example in soil properties. Possible sources of uncertainty and the way Eurocode 7 accounts for these uncertainties will be discussed. Basic principles of reliability-based design and probabilistic methods will be addressed. Finally, basic theory of retaining structures is discussed. 2.1 Design standard Eurocode 7 Eurocode 7 (Part 1: General Rules, Part 2: Ground investigation and testing) is the European Standard for the design of geotechnical structures. This design standard is based on a Limit State design approach in combination with a partial factor methodology. Goal of Eurocode 7 is to harmonize the design procedure of geotechnical structures and the evaluation of test results with respect to the selection of characteristic soil properties. The development of Eurocode 7 has been described in detail by Orr (2008) and is therefore not discussed here. In this section relevant parts of Eurocode 7 are highlighted, such as the basic principles of partial factors, design approaches and the use of finite element methods Serviceability limit state and Ultimate limit state Eurocode 7 exists as a guidance for geotechnical engineers and primarily describes the general principles and requirements for geotechnical designs in order to guarantee safety, serviceability and durability of structures. Traditionally lumped safety factors were applied to account for uncertainties in the design. In the past much experience was gained with this method and satisfactory designs were obtained. However, it is not possible to adequately account for uncertainties individually. Limit state design was introduced to design structures to the concept of limit states. These limit states have specific conditions that separate satisfactory and unsatisfactory states of a structure. In essence two limit states are distinguished in Eurocode 7, namely serviceability limit states (SLS) and ultimate limit states (ULS). These limit states are defined in EN1990 (Basis of structural design) as follows: Serviceability limit states (SLS): states that correspond to conditions beyond which specified service requirements for a structure or structural member are no longer met (e.g. excessive settlement related to the intended use of the structure) Ultimate limit states (ULS): states associated with collapse or with other similar forms of structural failure (e.g. failure of the foundation due to insufficient bearing resistance). Where in serviceability limit states particularly the deformations of a structure are verified, there are several failure modes that need to be checked when ultimate limit states are considered. Eurocode 7 lists the following five ultimate limit states, each with a different set of partial factors, that the geotechnical engineer needs to verify: Loss of equilibrium of the structure or the ground, considered as a rigid body in which the strength of structural materials and the ground are insignificant in providing resistance (EQU) 6

15 Internal failure or excessive deformation of the structure or structural elements, including e.g. footings, piles or basement walls, in which the strength of structural members is significant in providing resistance (STR) Failure or excessive deformation of the ground, in which the strength of the soil or rock is significant in providing resistance (GEO) Loss of equilibrium of the structure or the ground due to uplift by water pressure or other vertical actions (UPL) Hydraulic heave, internal erosion and piping in the ground caused by hydraulic gradients (HYD) Partial factors Eurocode 7 uses a partial factor methodology in combination with a limit state design. The safety of geotechnical structures is ensured by applying partial factors on actions and material properties or resistances. In the case of serviceability limit states the partial factors are equal to unity and the calculations are performed with the characteristic values (X k ). For serviceability limit sates, the following inequality is checked: E k C k (2.1) where E k is the characteristic value of the effect of an action and C k is the limiting value of the effect. In serviceability limit states the criterion is often the amount of displacement. The characteristic value of the effect of an action may therefore be the occurring settlement for example, the limiting value of that effect is the maximum allowable deformation. The choice of the characteristic value, X k, is an important aspect of Eurocode 7 and is treated further in Section 2.2. For the verification of ultimate limit states, Eurocode 7 provides recommended partial factors. If desired, these partial factors can be adjusted by each country itself, when specified in the National Annex. The use of partial factors and limit states is an essential development with respect to traditional design standards, where lumped overall safety factors were used to guarantee safety. Although this traditional design philosophy proved to perform satisfactory, it is unable to adequately account for all uncertainties found in a geotechnical design. With the limit state approach engineers are forced to think more cautious and rational about possible failure modes, which leads to less conservative designs in the end. In ultimate limit state calculations the design values (X d ) of properties are used. Design values are obtained by factoring the characteristic value with a partial factor (γ m ), in order to achieve a certain reliability. Loads or the effect of actions (E d ) and resistances (R d ) are then calculated as: E d = γ m X k R d = R k /γ m (2.2) Ultimate limit states are then verified with the following inequality: E d R d (2.3) Even though the way factors are applied has changed with the introduction of partial factors, a similar safety level is achieved as with lumped overall safety factors. This can be illustrated with an example of the design of piles (Vrouwenvelder et al., 2012). In traditional standards relatively low factors were used for driven piles in the Netherlands, and higher factors for in-situ piles. This resulted in overall safety factors of for displacement piles and for non-displacement piles. A similar safety factor is obtained after all partial factors have been applied according to Eurocode 7. For driven piles a partial factor γ F ( ) is applied on actions, a resistance factor on compression 1.2, and the correlation factor (ξ 3 ) to derive characteristic values from test results ( ). Together these partial factors result in a safety factor of approximately 2.2 ( ). If uncertainty exists in geometrical variables, such as ground and water levels, it is more convenient to directly adjust the parameters in the initial geometry of the problem. Eurocode 7 states that when variations in geometrical parameters are not important, characteristic values can directly be used as design values. On the other hand, when variations are significant for the calculation, geometrical parameters should be adjusted. In contrary to soil properties, this is not done by Chapter 2.1 MSc Thesis S.P. Kamp 7

16 applying partial factors. Characteristic values of geometrical parameters should be increased or lowered directly, depending on what is more unfavourable. For every design the considered limit state, as described in Section 2.1.1, can be denoted as a so-called limit state function Z(X d1, X d2,..., X dn ). This limit state function indicates for which variables, X di, the limit state is exceeded and for which it is not. Examples of more specific limit state functions will be discussed in Section 2.3. If Z(X d1, X d2,..., X dn ) is the limit state, a structure is considered safe enough when the following reliability function is fulfilled: Z(X d1, X d2,..., X dn ) > 0 (2.4) The partial factor methodology of Eurocode 7 has some obvious benefits, but it has been proven that it is difficult to reach agreement on how and when to apply the partial factors. Nowadays engineers often still apply only one single factor to make the transition from serviceability limit state to ultimate limit state in the case of complex geotechnical designs. Simpson (2000) argues that partial factors should be applied as directly as possible to uncertainties and in most cases they should be applied to soil strength rather than to resistances derived from the strength. In the next section the use of partial factors will be further elaborated. Distinction favourable and unfavourable actions In Eurocode 7 the distinction is made between favourable (or stabilizing) and unfavourable (destabilizing) actions. Generally favourable actions are decreased by a partial factor smaller than unity (γ f < 1), and unfavourable actions are increased with a partial factor larger than unity (γ f > 1). However, for some geotechnical structures this distinction may be problematic. This problem can be clearly illustrated for a retaining wall, as shown in Figure 2.1. The soil on and around the structure can be divided into several soil clusters to show the difficulty in determining whether an action is favourable or unfavourable. The complexity to distinguish favourable and unfavourable actions and applying the correct partial factor, may lead to conservative or over-designed structures in the end. Moreover, the nature of actions may change from one phase to another (i.e. initially a favourable action in first phase and unfavourable action in a subsequent phase). Figure 2.1: Typical geotechnical structure with both favourable and unfavourable actions (Bond and Harris, 2008). As mentioned above, the soil on and around the retaining wall can be divided in separate clusters. Individual soil clusters can affect particular failure mechanisms in a different way. The following effects can be distinguished for the retaining wall: An increase in wall self-weight and backfill above the heel is unfavourable for bearing resistance, but favourable for sliding resistance An increase in friction angle and cohesion (adhesion) below the construction is favourable for both sliding resistance and bearing capacity 8 MSc Thesis S.P. Kamp Chapter 2.1

17 The surcharge left of virtual plane is favourable for the overturning moment and sliding resistance, but unfavourable for bearing resistance The surcharge right of virtual plane is unfavourable for the bearing resistance, sliding resistance and overturning moment. If a geotechnical action can be identified as both favourable and unfavourable, a reasonable decision must be made since it is not logical to treat an action as both. This problem is stated in Eurocode 7 as the single-source principle : unfavourable (or destabilizing) and favourable (or stabilizing) permanent actions may be considered as coming from a single source in some situations. In this case the same single partial factor can be applied to the sum of all actions, although some actions may be favourable or unfavourable. Risk and reliability In order to achieve safe and reliable geotechnical designs, Eurocode 7 defines three Geotechnical Categories based on the complexity of the design. For each category the minimal requirements for the extent and content of geotechnical investigations, calculations and construction control are given together with the corresponding risks. There is no difference in the magnitude of partial factors for different geotechnical categories, but the categories indicate the required level of attention for investigations and calculations. Use of these categories is not mandatory and they are only provided as a guidance to identify the complexity and risks of the structure. Orr (2012) has treated this subject into more detail. Table 2.1: Definition of consequence classes and their corresponding target reliability index (β T ) (Gulvanessian et al., 2012). Consequence class CC1 CC2 CC3 Description Examples of buildings or civil engineering works Low consequence for loss of human life, or economic, social or environmental consequences small or negligible Agricultural buildings where people do not normally enter (e.g. storage buildings), greenhouses Medium consequence for loss of human life, economic, social or environmental consequences considerable Residential and office buildings, public buildings where consequences of failure are medium (e.g. an office building) High consequence for loss of human life and economic, social or environmental consequences very great Grandstands, public, buildings where consequences of failure are high (e.g. concert hall) Reliability class RC1 RC2 RC3 Reference period (years) β T More relevant in geotechnical engineering are the Consequence Classes and associated Reliability Classes (Table 2.1) as defined in Gulvanessian et al. (2012). Ultimate limit states concern both geotechnical and structural failure, and therefore also considers the risk to human life. The Consequence Classes show the consequences of failure to human life, economics and social or environmental consequences. Figure 2.2 shows the failure probabilities for serviceability limit states and ultimate limit states. General ranges of typical failure probabilities are indicated here. Geotechnical Categories can be compared with the Consequence Classes, but these classes involve target reliability indices (β T ). The reliability index takes the accepted or assumed statistical variability in action effects and resistances and model uncertainties into account. The required level of reliability is obtained by applying the recommended partial factors associated with a certain Reliability Class. The reliability index can also be easily noted in terms of the probability of failure. Further explanation on the importance of the reliability index is also found in Section 2.2. Chapter 2.1 MSc Thesis S.P. Kamp 9

18 Figure 2.2: Probabilities associated with limit states (Gulvanessian et al., 2012). Partial factors are calibrated with probabilistic methods on the basis of experience with traditional design methodologies, which use single safety factors, and statistical evaluation of experimental data and field observations. These factors are chosen in such a way that they provide similar designs as with lumped safety factors. Partial factors are independent of soil properties and the governing zone of influence. Therefore, the characteristic value (X k ) is the only parameter which has to completely describe the behaviour of the soil for a given design situation (Schneider and Schneider, 2012). The choice of an appropriate characteristic value will be discussed in Section 2.2. Vrouwenvelder et al. (2012) presented how the partial factors for the Netherlands have been calibrated with the use of probabilistic analyses. Various probabilistic methods and their application are discussed in Section 2.3 and Appendix C. The Dutch partial factors, as defined in NEN Annex A, are calibrated with the following formula: γ m = COV 1 α β COV (2.5) in which α is the influence coefficient, β is the reliability index and COV is the coefficient of variation. Equation 2.5 is only valid for normal distributed variables where the characteristic value has been taken as the 5% fractile. The coefficient of variation is measure of the variability of a parameter and can be obtained from soil investigation. It is concluded that the Dutch approach assigns higher partial factors to the most uncertain parameters with the highest influence on the design. For example, for sheet pile walls this resulted in a higher partial factor for the effective cohesion than the friction angle. It is also shown that a difference in Consequence Class leads to a different load factor, which is given in EN1990, for piled or raft foundations. In the case of retaining structures or slopes, this difference can be only taken into account by varying partial factors Design Approaches Eurocode 7 identifies three different Design Approaches to apply partial factors in geotechnical designs. The different approaches are a result of the different soil conditions and design traditions throughout Europe. Each design approach differs in the way partial factors should be applied to soil strength, resistances and actions. Although Eurocode 7 is drafted with the purpose to harmonize design standards, the choice of the design approach is determined by each country itself and then defined in the National Annex. The magnitude of partial factors is also determined nationally and must be specified in the National Annex. Often the chosen design approach is similar to the previously adopted approach in the national standard. Another difficulty of Eurocode 7 is that no clear guidance exists on the use of design approaches and partial factors in finite element methods. Different suggestions on how to apply partial factors in finite element methods are discussed in the next section. 10 MSc Thesis S.P. Kamp Chapter 2.1

19 Design Approaches provide a guidance in the transition from the serviceability limit state to the ultimate limit state. One of the difficulties with ultimate limit state design, both with deterministic methods and finite element methods, is how and when to apply these partial factors. Often serviceability limit state calculations are performed and the results (e.g. bending moments) are then multiplied with a certain factor to guarantee safety and to obtain the ultimate limit state results. For each type of geotechnical structure different partial factors are defined, which are listed in Appendix A of the Eurocode. Eurocode 7 introduces three design approaches DA1, DA2 and DA3 for the ultimate limit state checks of geotechnical and structural failure where the soil strength plays a significant role in providing resistance. Figure 2.3 shows the distribution of adopted Design Approaches throughout Europe for shallow foundations. More information on Design Approaches can be, among others, found in Eurocode 7 and Schweiger (2014). DA1: This approach requires two design combinations (DC) to be examined DC1: Factors on actions (loads) (DA1/1) DC2: Factors on material strength (DA1/2) DA2: Factors on actions and resistances DA3: Factors on actions and material strength. Figure 2.3: Design Approaches adopted by different countries for design of shallow foundations. (Bond, 2012) Eurocode 7 and Finite Element Methods Finite element methods (FEM) were initially developed to model stresses and deformations of geotechnical structures. Over recent years it has been proven to be a powerful tool to check serviceability limit states. However, the use of FEM for ultimate limit state design, e.g. checking against geotechnical failure, is more difficult. Since ultimate limit states require the application of partial factors the output may be altered unrealistically. Finite element methods are particular useful in cases where instability includes failure of both the Chapter 2.1 MSc Thesis S.P. Kamp 11

20 soil and structural members, such as failure surfaces that intersect retaining walls or piles. In case of these type of structures it is important to consider soil-structure interaction, which involves a difference in the relative stiffness. Soil-structure interaction effects are investigated more accurately with numerical methods, which include deformation analysis, than with traditional simplified analyses of ultimate limit states. Another advantage of FEM is that the failure mechanism does not have to be determined prior to the analysis, but is identified automatically. Several suggestions are made on how to apply partial factors and include the design approaches of Eurocode 7. How to apply partial factors in FEM There is still little experience with the use of partial factors and different Design Approaches with numerical methods. It has been shown by Schweiger (2014) that the concept of partial factors can be used for ultimate limit state design, but differences should be expected depending on the used design approaches. Therefore, more experience is required in performing ultimate limit state analysis in FEM. Checking several design approaches may be useful since weaknesses in the design may be revealed. The use of finite element methods for geotechnical verifications and the application of partial factors is investigated by Heibaum and Herten (2009), Schweiger (2014), Andrew (2012) and others. Heibaum and Herten (2009) show that FEM proves to be a reliable method for geotechnical designs. Especially in the case of complex geometries and construction processes, FEM proves to provide more satisfactory results than other methods. In the verification of serviceability limit states, characteristic properties are used. In order to check ultimate limit states, design values should be used. As explained above the design values are obtained by factoring the characteristic values with the partial factors defined in Eurocode 7. There are two different ways, which are similar to the design approaches, of introducing partial factors in FEM. The two limit state factoring approaches to check ultimate limit states in FEM are the Material Factoring Approach (MFA) and Load and Resistance Factoring Approach (LRFA). More information and examples on these methods are provided by Bauduin et al. (2000) and Andrew (2012). Material Factoring Approach (MFA) The Material Factoring Approach, also called Input Factoring, is more straightforward than the Load and Resistance Factoring Approach since input parameters are factored instead of output parameters. With MFA the critical failure mechanism is identified, as well as the margin of safety. This approach can be used for all types of problems generally, especially when the equilibrium of a soil mass is considered and checked against geotechnical failure. Since input parameters are factored directly at the source of uncertainty, it is a convenient approach to obtain design values of structural forces in the structural elements. However, it is important to pay attention to the effect of an action, since this affects how it should be factored. For example, the action of a column on a foundation is identified as a structural action, but the action applied by the foundation on a nearby retaining wall is a geotechnical action. Factoring soil properties has the effect that geotechnical resistances are reduced, which increases the amount of stress transferred to structural elements, usually causing structural forces to increase (Andrew, 2012). Bauduin et al. (2000) and Andrew (2012) state that factoring properties directly at the source as with MFA, is easier and more appropriate with FEM. This approach provides good results for checking geotechnical ultimate limit states. Satisfactory results are obtained for structural ultimate limit states. The Material Factoring Approach may be compared with Design Approach DA1/2 and DA3. Generally, Design Approach 3 is used in the Netherlands. Load and Resistance Factoring Approach (LRFA) With the Load and Resistance Factoring Approach, also called Output Factoring, the failure mechanism often has to be identified prior to the analysis. This may unnecessarily overcomplicate the analysis, since the governing failure mechanism is identified automatically with FEM. The design values of action effects and resistances are obtained by applying partial factors on the actions or the action effects and on the characteristic values of the resistances. The loading history is simulated with characteristic values of the strength parameters and actions. The design values of structural forces are obtained by multiplying the output values obtained with these characteristic values by the load factors. With complicated soil-structure interactions or consolidation and hardening behaviour, checking geotechnical soil failure is often difficult with LRFA (Schweiger, 2005). 12 MSc Thesis S.P. Kamp Chapter 2.1

21 The Load and Resistance Factoring Approach can be compared with design approach DA1/1 and DA2. Figure 2.4: Calculation schemes for MFA and LRFA (Bauduin et al., 2005). When to apply partial factors in FEM Besides the disagreement and lack of guidance on how to apply partial factors, there is also discussion on when to apply partial factors in the calculation process. Bauduin et al. (2000) proposed two schemes to perform ultimate limit state calculations in relation to serviceability limit state. It is argued that it would be more appropriate to start calculations with an initial stress field obtained with characteristic values. The safety at each stage can be verified by applying a so-called ϕ c reduction. The safety of the structure is guaranteed when strength parameters can be reduced until the value of the corresponding partial factor is reached. (a) (b) Figure 2.5: (a) Calculation Scheme 1 (b) Calculation Scheme 2 (Brinkgreve et al., 2016c). In the first scheme ULS calculations are performed separately for every SLS phase. Meaning that the first ULS phase (Phase 4) starts from the stress state resulting from the first SLS phase (Phase 1). This is then continued for every SLS phase, so Phase 5 (second ULS phase) starts from Phase 2 (second SLS phase) and so on. In Scheme 2 all ULS calculations are performed from Phase Chapter 2.1 MSc Thesis S.P. Kamp 13

22 1 (initial phase), meaning that Phase 4 starts from the stress state resulting from Phase 1, and Phase 5 starts from Phase 4. For calculations that finish successfully, e.g. without failure, can be regarded as calculations that fulfil the requirements of the design standard. In CUR166 (Guideline for Sheet Pile Walls) two similar schemes to Bauduin et al. (2000) are proposed. These schemes are also presented as a guidance to calculate ultimate limit states with finite element methods. CUR166 provides several steps in order to obtain a safe design in the ultimate limit state. Examples of important sheet pile wall elements to be checked are the embedded depth, moments, shear and normal forces, anchor forces and deformations. These example checks for sheet pile walls provide an appropriate indication of elements to be checked for general geotechnical applications. An important soil property for these types of structures is the soil stiffness. For the determination of soil stiffness, the following is mentioned: different stiffness moduli are preferably determined from in-situ and laboratory research, combined with experience. By means of statistical analysis a lower and upper bound are derived. Often it is not possible to determine these values for all layers, especially for more advanced constitutive models with two or three stiffness parameters. Calculation scheme 1 (CUR166): calculation with design values In this scheme, which can be compared with Scheme 2 of Bauduin, the calculations are performed with design values of soil parameters, embedded height, water levels and structural stiffnesses. For every case two ULS calculations are required, one with a lower bound design value and another with the upper bound design value of the soil stiffness. Besides ULS calculations, an SLS calculation is required to check for deformations. When the construction is in equilibrium until the last step, the embedded depth is sufficient. After equilibrium is guaranteed, all forces following from the calculation need to be checked. An advantage of this scheme is that it is relatively less labor-intensive than Scheme 2. Disadvantage is the overestimation of displacements in every construction phase, which leads to conservative forces and moments in the construction. Therefore, using Scheme 1 may not lead to the most optimal design. Calculation scheme 2 (CUR166): Calculation with characteristic values Scheme 2, which can be compared with Scheme 1 of Bauduin, uses characteristic values for the strength parameters (c and ϕ ). Design values are used for the embedded height, external loads and stiffness of soil and structures. This approach may be favourable in some cases, since geometrical parameters are not influenced by the ϕ c reduction. A calculation with the characteristic value of geometrical parameters would provide a unrealistic high safety level. As in Scheme 1, calculations are performed with a lower and upper bound value of the soil stiffness. At the end of each governing phase an ϕ c reduction should be performed to asses the safety. This should be done for each phase, since it may be difficult to identify which phases are governing in advance. In comparison with Scheme 1, this scheme provides a better insight in the occurring displacements, because characteristic values are used for soil strength parameters. Besides, a more optimal design is obtained due to the ϕ c reduction subsequently of each governing phase. The computed forces and bending moments are less conservative and therefore more realistic than with Scheme 1. In order to check the embedded depth, bending moments, shear forces and anchor forces the values, that are obtained after the reduction, need to be used. The ϕ c reduction should be continued until the factor is: 1.15 in safety class II 1.20 in safety class III 14 MSc Thesis S.P. Kamp Chapter 2.1

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24 2.2 Uncertainty It is generally known that geotechnical designs are significantly influenced by several uncertainties. The most significant uncertainty in geotechnical engineering is the result of the spatial variability of soil. With traditional deterministic design approaches it was not possible to take these uncertainties efficiently into account. Therefore, it is impossible to properly assess the reliability of a system. With the development of finite element analysis and other advanced programs it has become possible to include uncertainties and make a better assessment of the reliability of a structure. Reliability-based design is discussed in Section 2.3. In this section various uncertainties involved in geotechnical design, and their effects, will be elaborated Sources of uncertainty Dealing with uncertainty is a relevant aspect of geotechnical engineering. It is important to identify all uncertainties and how they affect the design. The overall uncertainty of geotechnical parameters is complex and results from many sources. Kulhawy (1993) distinguished three primary geotechnical sources of uncertainty that can be distinguished (Figure 2.6): inherent variability (aleatory): primarily caused by natural geologic processes that continuously modify in-situ soil transformation uncertainty (epistemic): introduced when field or laboratory measurements are transformed into input parameters for the design. Uncertainty due to simplifications and idealisations. measurement error: caused by sampling and laboratory testing. This error is increased by statistical uncertainty that results from the limited amount of information. Figure 2.6: Uncertainty in soil property estimates (Kulhawy, 1993). Most relevant categories are aleatory and epistemic uncertainty, which coincide in most geotechnical practical applications (Baecher and Christian, 2005). The third category is due to human errors and it is not covered here. The aleatory uncertainty is associated with the natural variability of soil properties, e.g the spatial variation of cohesion. This inherent uncertainty is often modelled with random variables and can be quantified by soil investigation measurements, statistical approximations and experience. Aleatory uncertainty cannot be reduced by gathering more data or by using more advanced models. Epistemic uncertainty is the consequence of the lack of knowledge. This uncertainty is related to limited data, measurement errors, incomplete knowledge, imperfect models and subjective judgement. This type of uncertainty can be reduced by collecting more experimental data, improving the measurements and calculation methods and by using more refined models. Furthermore, besides uncertainties introduced by the soil itself, another source of uncertainty is the model uncertainty. This is due to mathematical approximations and simplified statistical and physical models that are generally adopted (Honjo, 2011). Many tests and experiments in 16 MSc Thesis S.P. Kamp Chapter 2.2

25 geotechnical engineering on a real structure scale are available on retaining structures, which makes it possible to evaluate model errors. Honjo (2011) provides some examples on stability problem cases where this is shown. Even though this uncertainty source receives the least attention, it is shown that its magnitude is sometimes larger than the epistemic and aleatory uncertainty Coefficient of variation As described above, the uncertainty can be associated to three main sources in geotechnical engineering. In this research only inherent soil variability is considered, since probabilistic analyses can be used to take this type of variability into account. With these analyses uncertain variables are introduced as stochastic variables, described by their mean (µ) and standard deviation (σ). These statistics are then used to introduce another statistical parameter, namely the coefficient of variation (COV ). The coefficient of variation is defined as the ratio of the standard deviation over the mean of a parameter (COV = σ/µ). This non-dimensional variable describes if the scatter of a random variable is small or large relative to the mean value. It is considered the most straightforward and widely used parameter to describe the uncertainty of soil properties. The COV is used to make an approximation of the range of occurring soil properties. In optimal geotechnical design situations the coefficient of variation can be based on the variability provided by sufficient soil investigation data. Unfortunately there is often a lack of soil investigation. When this data is unavailable, estimations on the range of COV can be obtained from literature and design standards. Since the coefficient of variation is an useful and relatively simple parameter to model uncertainty, researchers studied typical values of this parameter and the effect on the design (Phoon and Kulhawy (1999), Phoon (2012), Hicks (2012)). NEN also provides values for COV s, which are shown in Table 2.2. Comparing the different proposed COV s, it can be seen that the values given in Table 2b of NEN are approximately the average of the range found in literature. Therefore it seems to be an obvious choice to use these values, however one should remember that the magnitude of COV may have a significant influence on the results. Table 2.2: Typical values of the coefficient of variation (COV ) from Orr (2008) and NEN Table 2b. Soil property Symbol Typical COV COV limited tests COV NEN Unit Volumetric weight γ [kn/m 3 ] Young s Modulus E [MPa] Friction angle ϕ [ ] Cohesion c [kpa] The coefficient of variation of a certain probability density function is derived from the mean and standard deviation from that particular distribution. It is important to choose the probability distribution appropriately, since assuming the wrong distribution may lead to unrealistic values for soil properties in some cases. Since one of the characteristics of a normal distribution is that the boundaries are minus and plus infinite, a small probability exists that negative values occur. Although this probability is extremely small in most cases, it may cause numerical problems in the analysis. In order to overcome this problem a lognormal, or another non-negative distribution, should be used. It is possible to use the lognormal distribution in any case, but the normal distribution is much easier for practical applications (Schneider and Schneider, 2012). Above described problems especially occur for high COV (COV 0.3). An example of a soil property where this often occurs is the cohesion or undrained shear strength. For high COV, a lognormal distribution should be used, since there is the probability of negative values (Schneider and Schneider, 2012). However, the probability of negative soil properties is insignificantly low in most cases. Due to variance reduction in the case of large soil volumes, it is often still acceptable to assume a normal distribution. For small coefficients of variation (COV < ) the results are very similar for normal and lognormal distributions, and the probability of negative values is close to zero. Chapter 2.2 MSc Thesis S.P. Kamp 17

26 2.2.3 Characteristic soil properties The characteristic value (X k ) is considered to be one of the most fundamental parameters for the verification of geotechnical structures in Eurocode 7. This parameter is used in the calculation of the serviceability limit state, and is required to calculate the design value (X d ) which is used in the ultimate limit state. The design value is obtained by dividing the characteristic value by the partial factor (γ m ). Eurocode 7 states that the design value is given by: X d = X k γ m (2.6) Since the partial factor is defined in the design standard, uncertainty in parameters is only taken into account by selecting the characteristic values appropriately. Eurocode 7 states that characteristic values should be selected with the aim that the probability of a more negative characteristic value governing the behaviour of the soil is smaller than 5% (Figure 2.7). Besides, it should be a cautious estimate of the value affecting the occurrence of the limit state. However, there has been some discussion on the term cautious estimate and the lack of clear guidance on how characteristic values should be derived. Evaluating the definition in Eurocode 7, it becomes clear that there exists no single definition for characteristic value. Several characteristic values exist, depending on the considered limit state. Attention should be paid to the definition of a characteristic parameter from a physical viewpoint and the definition of a characteristic value from a statistical viewpoint. The definition of a lower 5% fractile as a characteristic value must be interpreted as the lower 5% fractile of a probability distribution function which belongs to the design parameters, similarly defined based on the physics of the problem. In statistical terms the characteristic value is defined as: X k = µ X ± k n σ X = µ X (1 ± k n COV X ) (2.7) where µ X is the mean of the parameter X, σ X is the standard deviation of X, and k n is a statistical coefficient that depends on the number of samples n. In Equation 2.7, the magnitude of k n is for the 5% fractile with n =. Figure 2.7: Normal probabilistic distribution function for soil properties (Bond and Harris, 2008). Figure 2.7 shows a probability density function which is obtained by performing sufficient soil investigation, e.g. field and laboratory tests. In practice often insufficient soil data is available to compute a reliable probability density function. In that case the data of geotechnical parameters can be updated with the use of existing data or engineering judgement. Updating data in order to decrease the magnitude of uncertainty is called Bayesian updating and is explained in more detail by Orr (2000). The lower (or inferior) characteristic value (X k,inf ) is defined as the value of X below which 5% of all results are expected to occur (or a 95% probability that X will be greater than X k,inf. This value is used in situations where an overestimation of the magnitude of a material property may 18 MSc Thesis S.P. Kamp Chapter 2.2

27 be unsafe. On the other hand, the upper (or superior) characteristic value (X k,sup ) is defined as the value of X above which 5% of all results are expected to occur. The upper characteristic value is relevant in situations where underestimating the magnitude of a material property can be negatively. For example, for the soil weight behind a retaining wall an upper value should be chosen, since the force acting on the retaining wall largely depends on this weight. However, one can also argue that this property can be relatively easily determined and a less cautious value can be used. Due to the spatial variability of soil, characteristic soil properties are shown to be problemdependent (Hicks, 2012). Two competing factors exist due to this variability. First, the coefficient of variation is reduced due to spatial averaging of properties along the potential failure surface. Secondly, the tendency for failure surfaces to follow the path of least resistance, which causes an apparent reduction in the mean. The effect of spatial averaging is explained in more detail in the next section. When previously mentioned effects are not taken into account, it will probably lead to an over-conservative value of X k. Figure 2.8 shows the distribution of soil property X (the lower normal distribution). In order to satisfy the 95% reliability required in the Eurocode 7, the distribution of X has to be modified. To account for the soil heterogeneity it is more narrow and shifted to the left. Figure 2.8: General definition of X k (Hicks, 2012). Relevant volume of the soil (spatial averaging) The choice of the characteristic value of soil properties is also influenced by the relevant soil volume of the problem. The relevant volume influencing the problem differs for various types of geotechnical applications. For example, in a slope stability problem which involves a large volume of soil, stability is governed by the shear strength average along the failure surface. Since it involves a large volume of soil in this case, the failure surface probably passes through both weak and strong zones. In the case of a single foundation pile, the failure involves a much smaller soil volume. If the pile is founded in a weak zone of the same soil as the slope, the mean governing value will be lower. These examples show that it is important to make a cautious estimation of the characteristic value for each particular application, i.e. the characteristic value for the pile will be lower (more cautious) than for the slope. Hicks (2012) listed extracts from Eurocode 7 where the important parameters in selecting the characteristic value are mentioned. One of those extracts mentions the extent of zone of ground governing the behaviour of the geotechnical as an important parameter, which indicates the spatial characteristic and importance of variability for the characteristic value. Eurocode 7 also implies that the characteristic value is problem-dependent with the ability of geotechnical structures to transfer loads from weak to strong zones in the ground. In reality the soil volume governing the considered limit state is often large relative to the size of investigated soil tests. These tests only consider a small volume of the real problem volume, which may provide a wrong and distorted representation of the actual soil strength. However, Eurocode 7 states the value of the governing parameter is often the mean of a range of values covering a large volume of ground. Chapter 2.2 MSc Thesis S.P. Kamp 19

28 It is generally shown that it is important to include the spatial nature of soil variability. This type of variability is represented by the scale of fluctuation (θ), which is a measure of the distance over which property values are significantly correlated (Hicks, 2012). Since it is not possible to include the scale of fluctuation in this thesis, it is not discussed extensively here. However, some convenient and interesting features are worth noting. When the scale of fluctuation would be included in the model, it is the domain size relative to the scale of fluctuation that is important in assessing the relevance of the mean property value. Not the domain size itself, or its size relation to test samples and field test although this may have an influence. It may be reasonable to choose the mean property value over a potential failure surface, but this value may differ significantly from the mean when the entire soil volume is considered. This is due the tendency of failure mechanisms to follow the path of least resistance. The difficulty is to identify the potential failure mechanisms in advance, and to what extent these mechanisms and characteristic values are influenced by the spatial structure of heterogeneity. Figure 2.9: Characteristic values for different design situations (Orr, 2000). Orr (2000) illustrates the problem with the selection of the characteristic value, and knowing how cautious it should be, from data obtained by tests. In Figure 2.9, undrained shear strength (c u ) values obtained from tests on clay are plotted. This problem clearly shows the difficulty in choosing an appropriate characteristic value. The solid line is the average value of the test results, c uav, obtained from a regression analysis. The dotted line is the cautious estimate of these average values if a failure mechanisms passes through all of the soil. This cautious estimate is based on experience and the spread of data and is the same as the design value that was selected traditionally. It is found the characteristic value is obtained using Schneider s method, which is defined as the mean minus one half of a standard deviation. The dashed line represents the case when the failure mechanism only passes a small volume of the ground (e.g. base resistance of a pile). When the failure mechanism considers a relatively small volume of soil the characteristic value should be chosen more cautious. The line that represents the cautious estimate can be seen as a lower bound to the results. Effect characteristic value From the previous section it became clear that it is problematic to determine the correct characteristic value for a particular problem. Several definitions of the characteristic value in are presented in literature. In order to show the importance of an appropriate chosen characteristic value, the effect of various characteristic value definitions are investigated by Marques et al. (2011). In this research a retaining wall is analysed for eight sets of characteristic soil properties (X k ): mean values (1), Schneiders equation (2), Ovesens equation (3), 95% reliable mean values (for unknown or known COV, referenced as mean) (4), 5% fractile values (from normal distribution) (5), and 5% 20 MSc Thesis S.P. Kamp Chapter 2.2

29 fractile values (according to the number of test results and for unknown or known COV, referenced as low) (6): (1) X k = X m (4) X k = X m (1 k n,mean COV ) (2) X k = X m 0.5σ X (5) X k = X m 1.645σ X (3) X k = X m 1.645σ X / N (6) X k = X m (1 k n,low COV ) in which X m is the mean value, σ X is the standard deviation, COV is the coefficient of variation, N is the number of test results, and k n,mean, and k n,low, are statistical coefficients taking into account the sampling, the number of test results, the value affecting the occurrence of the limit state, and the statistical level of confidence required for the assessed characteristic value, expressed by a considered t-factor of the Students distribution. In this study a relatively homogeneous soil is considered and the considered soil properties are assumed to be normally distributed. Figure 2.10: Analysed concrete retaining structure (Marques et al., 2011). Marques et al. (2011) investigated the differences between the definitions of the characteristic value described above. Besides, also the influence on the reliability index for different design approaches is investigated. In this example a concrete retaining structure is analysed, which is shown in Figure The influence of using different definitions for the magnitude of characteristic soil parameters is shown in Figure 2.11, where ϕ w is the friction angle of the soil on the active and passive side of the wall, γ is the unit weight of the soil, c f is the cohesion, ϕ f is the friction angle of the foundation soil. It is shown that adopting different definitions to determine the characteristic value, may result in a wide spread in magnitude of a soil property. For example, the cohesion varies largely between almost 15 kp a and about 5 kp a. Therefore, the characteristic value should be chosen accordingly to the problem. The influence of different definitions of characteristic values on the reliability index are also presented (Figure 2.12). The reliability index is obtained by the First Order Reliability Method (FORM) for eight definitions of the characteristic value, considering c f normal or lognormal and uncorrelated random variables for all Design Approaches (Figure 2.12). The target reliability index in the ultimate limit state for this problem is β T = 3.8. If, for example, c f is assumed normal and DA3 is applied, it can be seen that the reliability index varies between a minimum of about 3.5 and a maximum of 7. Since in two cases the required reliability index is not met, it is shown that is necessary to choose the characteristic value appropriately to the problem. Chapter 2.2 MSc Thesis S.P. Kamp 21

30 Figure 2.11: Characteristic values of soil properties X k based on the definition of eight sets (Marques et al., 2011). Figure 2.12: Reliability index β based on the definition of eight sets (Marques et al., 2011). 22 MSc Thesis S.P. Kamp Chapter 2.2

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32 2.3 Reliability-based design Traditionally, the safety of geotechnical structures is based on an overall factor of safety. An alternative to this method is the partial factor based approach as adopted in Eurocode 7 (Section 2.1). Eurocode 7 uses characteristic values and partial factors to ensure a safe design. As discussed in Section 2.2 many sources of uncertainty exist in geotechnical engineering. Reliability-based methods explicitly account for uncertainty of input parameters with the use of stochastic parameters. These stochastic parameters are quantified by statistical analysis of soil investigation data, experience and engineering judgement. The term reliability is often defined as: the ability of a structure or structural element to fulfil the specified requirements, including the working life, for which it has been designed. Reliability-based methods provide a more meaningful way of evaluating the influence of uncertainties in the design. However, reliability-based methods are not often used in engineering practice. Due to the fact that reliablity-based methods generally require more input data, which is often not available, they are not regularly applied in engineering practice. Besides the computational effort can be significantly higher with these methods. At this time reliability-based methods are primarily used for research in geotechnical engineering. Commonly used methods, such as Monte Carlo simulation, First Order Reliability Method and Point Estimate Method, are documented in detail by Griffiths and Fenton (2007). Goal of adopting probabilistic methods is to harmonize geotechnical design. Phoon and Ching (2014) mention that reliability-based methods should not replace the traditional factor of safety, but that they should be used together. It is suggested that it only takes a little more effort with some methods to obtain the reliability index and probability of failure of a geotechnical structure. Main purpose of reliability-based design compared to deterministic methods is to model the heterogeneity of the soil. Basically, heterogeneity can be represented in two ways: different layers or areas are specified; no specific layers are modelled, but the spatial variability is modelled In the second case, it is necessary to implement random field theory (Griffiths and Fenton, 2007). Random field theory is the most sophisticated way to model uncertainty, but is not considered here. The randomness of soil properties in the first case is modelled using probability theory. Soil properties are modelled as random variables by means of their distribution. In geotechnical engineering only the uncertainty in soil properties is considered generally. In this section the principles of reliability-based design will be discussed. Several methods to evaluate the reliability of geotechnical structures are explained and compared. It is also discussed how the input uncertainty can propagate through the model and affects the output Reliability analysis The partial factor method as in Eurocode 7 uses partial factors to guarantee safety. This method was originally first used in structural engineering. However, more sophisticated methods exist to assess the safety of geotechnical structures. With reliability-based methods safety is no longer necessarily expressed in terms of a factor of safety, but rather with failure probability (P f ) or reliability index (β). A more useful approach is the use of limit state theory to determine whether a structure can be considered safe. A construction is considered safe if the resistance (R) is larger than the load (S): R > S (2.8) 24 MSc Thesis S.P. Kamp Chapter 2.3

33 Figure 2.13: Probability of failure as probability of the load exceeding the resistance (Schweckendiek, 2006). When the distributions of both the resistance (R) and load (S) are known, the failure probability (P f ) can be determined. Failure occurs when R < S. The probability of failure equals P f = P [S < R], as can be seen in Figure An even more efficient way to denote the probability of failure is with the use of a limit state function (Z). The limit state can be seen as a condition beyond which the structure no longer fulfils a particular limit state function. The limit state (Z) can be considered in the following way: Z = R S (2.9) With this definition the probability of failure can be computed as P f = P [Z < 0]. More general the probability of failure is expressed through this limit state function (Equation 2.9). In some simple cases this equation can be solved by analytical integration, as is shown in Equation Other methods to compute the probability of failure are numerical integrations, approximate analytical methods (e.g. First-Order Reliability Methods) and simulation methods. P f = P rob(z 0) = ϕ(x)dx (2.10) If both the variables R and S are both assumed to be normally distributed and uncorrelated, Z also can be taken normally distributed. Fundamental probability theory then provides that the mean (µ) and standard deviation (σ) of R, S and Z are related by: Z 0 µ Z = µ R µ S (2.11) σ 2 Z = (σ 2 R + σ 2 S) (2.12) Obviously variables can also be correlated in reality. In this case the equation above needs to be slightly expanded. When both variables are correlated with the correlation coefficient (ρ RS ), the last equation is defined as: σ 2 Z = σ 2 R + σ 2 S 2ρ RS σ R σ S (2.13) As Figure 2.14 illustrates, the probability of failure is the area below the distribution of Z located the left of Z = 0. If S and R are both normally distributed, Z is normally distributed as well. Then it follows that the probability of failure P f is: P f = Φ( β) (2.14) where Φ is the cumulative distribution function (cdf) of the standard normal distribution and β is defined by Equation In terms of the factor of safety, the probability of failure is defined as the probability that the factor of safety is less than unity. Generally, according to the Eurocodes, a reliability index greater than β T = 3.8 must obtained for a 50 year reference (Gulvanessian et al., 2012). The probability of failure and the corresponding reliability index are only conceptual values Chapter 2.3 MSc Thesis S.P. Kamp 25

34 that not necessarily represent actual failure rates. They are used as operational values for code calibration and to compare reliability levels. Table 2.3: Relationship between the probability of failure (P f ) and the reliability index (β). P f β As mentioned before, the simple definition of the reliability index is based on the assumption that capacity and load are normally distributed and the limit state is the situation that their difference, the margin of safety Z, is zero. The random variable Z is then also normally distributed. The reliability index is a safety margin that indicates how many standard deviations (σ Z ) the mean value (µ Z ) of the performance function is removed from the critical limit state (Z = 0). When uncorrelated, normal distributed variables are used, the reliability index is defined as follows: β = µ R µ S σ 2 R + σ 2 S = µ Z σ Z (2.15) In geotechnical engineering variables, e.g. cohesion, are sometimes assumed non-normal to prevent the occurrence of negative values. It is also possible that the output of the reliability-based design, e.g. bearing capacity, follows a non-normal distribution. Most commonly used non-normal distribution in geotechnical engineering is the lognormal distribution. For the lognormal distribution the definition of the reliability index becomes more complicated: β = m M s M = [ ] m ln Q 1+COVF 2 m F 1+COVQ 2 (2.16) ln[(1 + COVQ 2)(1 + COV F 2)] Figure 2.14: Distribution of limit state function Z (Gulvanessian et al., 2012) Reliability-based methods The reliability of a structure depends on the safety margin between the resistance to failure and the applied load. Various methods exist to describe and determine this margin of safety. These methods are divided into three categories: fully probabilistic, fully probabilistic with approximations and semi-probabilistic. In this section an overview of various probabilistic methods is given. 26 MSc Thesis S.P. Kamp Chapter 2.3

35 Figure 2.15: Overview reliability methods (Gulvanessian et al., 2012). Level III method (fully probabilistic) Level III methods include reliability methods that use the full probability density functions of all uncertain parameters without simplifications and approximations of the model. An example of a Level III reliability method is Crude Monte Carlo simulation (Appendix C). Level II method (fully probabilistic with approximations) Level II calculations contain several methods to determine the probability of failure and reliability index, where the performance function is approximated as a linear or second order function in the design point. When a linear function is approximated, the First Order Reliability Method is generally used (Appendix C). This method usually uses the first two moments of a uncertain variable, i.e. the mean and variance. A more straightforward method is the Point Estimate Method. The Point Estimate Method will be elaborated in the next section. Level I (semi-probabilistic) For Level I methods prior knowledge of the effects of uncertainties is required. These methods do not calculate a reliability, but only obtain a target reliability index (e.g. Eurocode 7). A calculation is done following a design standard, which considers a structure to be safe when there is a margin between the value of the strength and the loads. This margin is provided with the use of partial factors which are calibrated prior by probabilistic studies Point Estimate Method The previous section shortly listed several levels of reliability-based methods. A Level II probabilistic method is the Point Estimate Method (PEM), which will be used extensively in this research and will be elaborated in this section. The Point Estimate method is a relatively simple method to evaluate the reliability of a structure. This method is a computationally straightforward approach to explicitly account for uncertainty of input parameters. This method is able to estimate the statistical moments, i.e. mean and variance, of the output. The general idea is to simplify the entire distribution of a variable by a discrete equivalent distribution. This is done by assigning the same three first statistical from the complete original distribution to the new equivalent distribution. Before performing calculations with PEM, evaluations points need to be defined. Generally two evaluations points are defined, located at one standard deviation on either side of the mean value. This is done for each stochastic input parameter. Next the performance function is calculated for every possible combination of the evaluation points. This results in 2 n calculations, where n is the number of included stochastic variables. One disadvantage of PEM is that it does not provide a complete output distribution, as is the case with Monte Carlo. On the other hand, PEM requires Chapter 2.3 MSc Thesis S.P. Kamp 27

36 only little knowlegde about probability theory knowledge. Another advantage is that could be applied for any probability distribution. When the probability distribution function for the output variable is known, for example from previous Monte Carlo analyses, the mean and standard deviation values can be used to calculate the complete output distribution. Figure 2.16: Evaluation points on probability density function. The basis of the Point Estimate Method is developed by Rosenblueth (1975). Original Rosenblueth deals with three cases: 1. When Y is a function of one variable X, whose mean, variance, and skewness are known 2. When Y is a function of one variable X, whose distribution is symmetrical and approximately Gaussian 3. When Y is a function of n variables, X 1, X 2,..., X n, whose distributions are symmetric and which may be correlated. Commonly calculations are made at two evaluations points and Rosenblueth uses the following equation: E[Y ] P + y + + P y (2.17) where Y is a deterministic function of X, Y = G(X), E[Y m ] is the expected value of Y, y + is a value of Y evaluated at a point x + which is greater than the mean, y is a value of Y evaluated at a point x which is less than the mean, P + and P are the weights. Case 1 For the first case Rosenblueth s provides four conditions that must be satisfied for the first statistical moments: P + + P = 1 (2.18) P + x + + P x = µ x (2.19) P + (x + µ x ) 2 + P (x µ x ) 2 = σ 2 x (2.20) P + (x + µ x ) 3 + P (x µ x ) 3 = ν x σ 3 x (2.21) where ν x is the skewness of X. The solutions for these four conditions are: ] [ν ( x2 x ± = µ x + ± νx ) (2.22) P + = 1 2 [ 1 ν x 2 ] (νx /2) 2 (2.23) Often normal distributions are used, i.e. the skewness is zero, which simplifies the solutions. When the skewness is zero, the distribution of the variable X is symmetric, the solutions become: P + = P = 1 2 ; x + = µ x + σ x ; x = µ x σ x ; (2.24) Case 2 In Case 2, when the distribution X is symmetrical and approximately normal, the evaluation 28 MSc Thesis S.P. Kamp Chapter 2.3

37 points x can be estimated at more than two points. For three points this involves a central point at the mean µ x and two points at x + and x which are symmetrically distributed about the mean. The weight for the central point is defined as P and the other two notation stay the same, therefore: The solutions for these equations are: 2P + + P = 1 (2.25) 2P + (x + µ x ) 2 = σ 2 x (2.26) 2P + (x + µ x ) 4 = 3σ 3 x (2.27) P = 2 3 P + = P = 1 6 (2.28) x ± = µ x ± 3σ x (2.29) Case 3 Case 3 is the most widely used application of Rosenblueth s method. This case is a generalization of Case 1, when the skewness is ignored. In this procedure calculations are done at 2 n points, so that the value of each variable is at one standard deviation below or above the mean. For two variables, four calculations are required at the points (µ x1 σ x1, µ x2 σ x2 ), (µ x1 σ x1, µ x2 +σ x2 ), (µ x1 + σ x1, µ x2 σ x2 ), (µ x1 + σ x1, µ x2 + σ x2 ). If the variables are uncorrelated, the weight of each point is P i = Figure 2.17: Example evaluations points and weights of correlated soil properties. Input variables may also show a correlation. The principle of correlation is schematized for two correlated variables in Figure If two variables are correlated with the correlation coefficient ρ, the points are still located at one standard deviation below or above the mean, but the weights are adjusted. For the same evaluation points the weights become (1 + ρ)/4 and (1 ρ)/4, as shown in Figure 2.18 (left). PEM also allows more than two variables to be included as correlated, which is illustrated in Figure 2.18 (right). To obtain a clear definition of the weights, Rosenblueth used a set of + and as subscripts. When three correlated variables X are considered, the first sign refers to X 1, the second to X 2, and the third to X 3. The sign is positive when the evaluation point is considered one standard deviation above the mean value. For three correlated variables, where ρ 12 is the correlation coefficient between X 1 and X 2 and so on, the weights are defined as: P +++ = P = 1 8 (1 + ρ 12 + ρ 23 + ρ 31 ) P ++ = P + = 1 8 (1 + ρ 12 ρ 23 ρ 31 ) P + + = P + = 1 8 (1 ρ 12 ρ 23 + ρ 31 ) (2.30) P + = P ++ = 1 8 (1 ρ 12 + ρ 23 ρ 31 ) Chapter 2.3 MSc Thesis S.P. Kamp 29

38 Figure 2.18: Rosenblueth s points and weights for two (left) and three (right) variables (Rosenblueth, 1975). For a single random variable, the weights are given by: P x+ = ξ x /(ξ x+ + ξ x ) P x = 1 P x+ (2.31) When for two variables the evaluation points are both at the same side of the mean (e.g. both one standard deviation above the mean), the sign of the correlation coefficient is positive. If the location is opposite, the sign is negative. For n variables, 2 n points are required to include all possible combinations with each variable one standard deviation above or below the mean. With n variables, the weights are: P (s1s 2,...,s n) = 1 2 n 1 + n 1 i=1 j=i+1 n (s i )(s j )ρ ij (2.32) where s i is positive when the value of the ith variable is one standard deviation above the mean and negative when the value is one standard deviation below the mean, y i is the value of Y evaluated at x i, and i is an appropriate combination of + and signs indicating the location of x i. For uncorrelated variables Equation 2.32 reduces to P i = 1/2 n. For each input variable the evaluation locations are computed. The locations are determined with the use of so-called standard deviation units (ξ) (Griffiths et al., 2002). When a normal distribution is assumed, i.e. skewness is excluded (ν xi = 0), the standard deviation unit is equal to ξ = 1. For a random variable, with skewness (ν xi ), the standard deviation units are given by: ξ xi+ = ν ( ( x i 2 + νxi ) ) 2 1/2 1 + (2.33) 2 and ξ xi = ξ xi+ ν xi (2.34) where the subscripts and + represent whether the evaluation point is located below or above the mean respectively. As mentioned above, for a symmetrical normal distribution the standard deviation units are equal to unity. The locations of the evaluation points are then calculated with: x i+ = µ xi + ξ xi+ σ xi (2.35) x i = µ xi ξ xi σ xi (2.36) When skewness is taken intro account, e.g. for lognormal distributed variables, the skewness ν xi is calculated with use of the coefficient of variation (COV ) according to Equation 2.37 (Benjamin and Cornell, 2013). ν xi = 3 COV + COV 3 (2.37) 30 MSc Thesis S.P. Kamp Chapter 2.3

39 After all input variables are defined and the calculation has been performed, the output statistics can be computed. Typical output statistics of PEM are the mean (µ xi ), standard deviation (σ xi ) and possibly the skewness (ν xi ). The first moment, or µ xi, is calculated as follows: µ xi = 2 n i=1 P i y i (2.38) The second central moment, or variance σ 2 x i, is defined as: σ 2 x i = 2 n i=1 P i (y i µ) 2 (2.39) A more common quantification of the output deviation is the standard deviation, which is calculated as the square root of the variance: σ xi = 2n P i (y i µ) 2 (2.40) i=1 Possibly the skewness ν xi can be calculated, which is a measure of asymmetry of the probability density function. It has to be mentioned that it is suggested by Christian and Baecher (1999) that PEM should not be used for moments higher than the second moment. The skewness is defined as: ν xi = 1 n 2 σ 3 P i (y i µ) 2 (2.41) i= Output uncertainty With all the output statistics calculated as described above, it is possible to generate a probability density function. A disadvantage of PEM is that the exact output distribution shape is not known. Distributions of input variables can be determined from data obtained with the use of soil investigation, for example. However, it can not be stated that when all input variables are normal distributed, the output distribution of a certain parameter also follows a normal distribution. Due to the non-linearity of finite element models it is possible that the output distribution deviates significantly from a normal distribution. The exact shape can be determined with Monte Carlo, but this method is inconvenient for engineering practice due to the long computation times. Since the output distribution has to be assumed, it is desirable to have some knowledge about the propagation of input uncertainty through to model. Valley and Kaiser (2010) investigated the propagation of uncertainty in PEM for underground excavations. It is concluded that PEM proves to be an efficient method to include uncertainty. On the other hand the simplicity of PEM also comes with some limitations. Accurate results are obtained in the case of uncorrelated normal distributed variables. Generally the uncertainty is well captured by PEM, but is unable to correctly capture the behaviour of the structure in the tails. Even when all input parameters are taken normally distributed it is possible the distribution is inaccurate in the tails. This problem may be solved by adopting more advanced PEM variations that calculate higher-order moments, or using evaluation points further away from the central tendency (Che-Hao et al. (1995), Christian and Baecher (2002)). Chapter 2.3 MSc Thesis S.P. Kamp 31

40 Figure 2.19: Increasing uncertainty in the output distribution for increasing non-linearity compared to Monte Carlo computations (Valley and Kaiser, 2010). Another limitation of PEM is encountered when both elastic and plastic soil behaviour is involved the finite element analysis. When this occurs the output distribution may not be captured accurately, as can be observed in Figure Valley and Kaiser (2010) recommend to test the effect of this phenomenon by using a limited number of stochastic parameters at a time. Using different stochastic parameters shows how the output is affected and allows to get a better understanding of the possible outcomes. Besides it also provides a better insight on the potential error introduced by the this method. In addition to the approach described above, deterministic calculations may provide a better understanding of the output distribution. Deterministic calculations with extreme values from the tails of input parameters can be performed. Solutions of these additional calculations can be compared with the tails of the output distribution to see if similar behaviour is captured. Accuracy Some significant assumptions are made in both the input and output of the Point Estimate method, which makes it difficult to determine the accuracy of this method. Christian and Baecher (1999) provides some guidance in the expected accuracy of Rosenblueth s method when it is used with two evaluations points. Gaussian quadrature is a numerical approximation to the integral: I = g(z) f(z) dz (2.42) The limits of this integral depend on the function that is integrated. In the equation above, g(z) is the function to be evaluated at the integration points, f(z) is the so-called weighting function. The integral can be approximated as follows: I H i g(z i ) (2.43) The magnitude of the weigts (H i ) are determined by the function f(z). The locations of the integration points and the values of H i are determined by PEM, for every assumed distribution of the input variable X. Gaussian quadrature with n points then guarantees that the integration will be exact for a polynomial function g(x) of an order up to and including (2n 1) variables. In the case of the two-point PEM, the results will be exact for third-order polynomials or less. Suppose 32 MSc Thesis S.P. Kamp Chapter 2.3

41 Y is a linear function of the variable X, then the variance of Y will be of order two, and PEM will compute both the mean and the variance of Y exactly. If Y is a function of X 2, the mean will be computed exactly, but the variance will be of order 4, and some error will occur (Figure 2.20). Figure 2.20: Error in estimate of variance for Y = X 2, X normally distributed (Christian and Baecher, 1999). More examples are shown in Christian and Baecher (1999). Generally, the error is acceptable for small COV, but will increase for an increasing COV. It is shown that in some cases the method requires more than two evaluation points to obtain reasonably accurate results. This method performs best when small COV s are used and the functions of X can be integrated approximated reasonably by a third-order polynomial. Also the results obtained by Christian and Baecher (1999) can be used to estimate errors. It is also stated that results from truncated Taylor series, such as First Order Second Moment methods, are generally less accurate, because they are based on lower order expansions. Output distributions and limit state functions The Point Estimate Method does not provide the exact shape of the output distribution, only a number of values that are assumed to belong to a certain distribution. For simplicity, normal distributions can be assumed for output parameters, such as displacements, bending moments and the factor of safety. However, this is not always the case in reality.in some cases a lognormal, or other distribution shape, proves to be a better option. Some general insight in the output distributions of common output parameters is provided here. Without going into too much detail of the investigated structures, it is found that the output distributions can vary widely. Chew et al. (2015) shows the output distribution, for different values of the COV, for a shallow foundation vary from clearly lognormal to approximately a normal distribution. Zevgolis and Bourdeau (2010) investigated the factor of safety (and reliability index) of different failure modes of a retaining wall, which all show to follow a lognormal distribution. The same is shown by Guha Ray et al. (2014), where the reliability of a retaining wall against sliding is considered. The factor of safety is best-fitted with a lognormal distribution, while a normal distribution is obviously less appropriate. In the case of slope stability problem (Hicks and Nuttall, 2012), where soil variability is accounted for with random fields, shows that a normal distribution fits the factor of safety reasonably well. In geotechnical engineering it is usual, but conservative, to assume a normal distribution for the factor of safety (Baecher and Christian, 2005). When a normal distribution is assumed, the reliability index is computed as (Phoon and Ching, 2014): β normal = µ F S 1 (2.44) σ F S The nominator in Equation 2.44 is essentially the limit state function as discussed in Section Generally a structure is considered safe if the factor of safety is larger than unity. Therefore the top part of the fraction is the margin between failure and non-failure. When this margin is divided Chapter 2.3 MSc Thesis S.P. Kamp 33

42 by the standard deviation of the output distribution, the reliability index (β) is obtained. More useful is a generalization of the equation above, which can for example be used for displacements or bending moments criteria. A limiting value (Z max ) of the examined criteria must be defined. Other parameters, µ Z and σ Z, are obtained from the output. A general definition of the reliability index, assuming a normal output distribution, is defined as follows: β = Z max µ Z σ Z = Z σ Z (2.45) When the reliability with respect to bending moments is assessed, for example in a sheet pile wall, the maximum bending moment is a function of the yield strength. More examples of limit state functions can be found in Schweckendiek (2006). Fractiles output distribution When the distribution of a parameter is known, the characteristic and design values can be calculated. The characteristic value is defined as the lower 5% fractile. For a normal distribution the fractiles are statistically defined as: X k = µ k σ = µ(1 k P,α COV ) (2.46) where the k P,α coefficient is equal to 1.64 for the 5% fractile. For lognormal distributions it is calculated as: µ X d = exp ) ( k P,α ln(1 + COV 2 ) 1 + COV 2 (2.47) This formula conveniently uses the same k P,α value as for normal distributions for the same fractile. For design values the same formulas as for characteristic values are used, only different k P,α coefficients are used. The design value is generally defined as the lower 0.1% fractile and its coefficient is equal to k P,α = In Table 2.4 also the k P,α values are shown when the skewness is included. If the skewness ν x is calculated, values from this table can be used in the formulas above. However, since the Point Estimate Method only provides little data, determining the skewness is associated with uncertainty. For a small skewness, ν x = 0.1, and a low coefficient of variation the normal distribution can be used as a good approximation. When the skewness is positive, the predicted lower fractiles from a lognormal distribution are generally greater (or favourable) than those obtained from a normal distribution. In the case of negative skewness, lower fractiles from a lognormal distribution are lower than for a normal distribution. When a normal distribution is used, and there is an actual negative skewness, the predicted lower fractiles will have an unfavourable error. In other words, they will be larger than the actual correct values. In the case of a positive skewness, the reverse is true. So when a normal distribution is used, and the actual distribution has a positive skewness, the obtained values will be conservative. It is recommended to consider skewness when the coefficient of variation is greater than COV = 0.1 (Gulvanessian et al., 2012). It is stated in Gulvanessian et al. (2012) that with a lack of actual data, i.e. a minimum sample size of 30, no credible value for the skewness can be determined. Table 2.4: Coefficient k P,α for the determination of the lower 5% and 0.1% fractiles, assuming a three-parameter log-normal distribution. Coefficient of skewness α X Coefficient k P,α for P = 5% Coefficient k P,α for P = 1% MSc Thesis S.P. Kamp Chapter 2.3

43 2.3.5 Comparison reliability-based methods The Point Estimate Method has been investigated in several studies (Schweiger et al. (2001), Russelli (2008), Chew et al. (2015), Russelli and Vermeer. (2005)). In these studies PEM has been compared with other probabilistic methods, such as Monte Carlo and FORM. Some general advantages and limitations of PEM can be concluded from different studies. Advantages of the Point Estimate Method: It is not necessary to know the exact shape of the probability distribution of the input variables Besides the mean and standard deviation, also the skewness coefficient can be computed with PEM. With little increase in computational effort, it can provide more accurate results than FORM With PEM it is not required to compute the derivatives of the performance function to obtain the mean and standard deviation of the output Since PEM is a non-iterative procedure, it does not have convergence problems like FORM (Russelli and Vermeer., 2005) Less computational effort for a comparable accuracy. Also for non-linear functions the behaviour is well captured (Russelli and Vermeer., 2005) In most applications to geotechnical engineering PEM has been used for uncorrelated variables. The additional effort required to include correlation is insignificant. One of the advantages of PEM is the ease with which it can be applied to problems with multiple stochastic variables (Christian and Baecher, 1999). Peschl and Schweiger (2003) state that the overall behaviour is fairly accurate and well captured for moderate magnitudes of variability in the soil (COV < 0.5) with PEM. Limitations of the Point Estimate Method: No information about the output probabilistic distribution is provided. The shape of the distribution has to be assumed, introducing additional uncertainty in the final results. However, the normal and lognormal distribution are commonly used in geotechnical engineering Two evaluation points may not be adequate to obtain accurate estimates of the moments. However, for most practical cases two evaluation points are sufficient. (Christian and Baecher, 1999) If more accuracy is desired, a larger number of random input variables is required. Also moments higher then the standard deviation have to be considered. This increases the amount of computational effort (Russelli and Vermeer., 2005) Christian and Baecher (1999) suggest that PEM should not be used for higher moments than the second moment for non-linear functions. They also conclude that the output error increases for larger COV of input variables. It must be noted that most practical reliability methods do not use moments higher than the second. The original Point Estimate Method for multiple input parameters requires 2 n calculations. When n is larger than 5 or 6, the number of calculations becomes to large for practical applications (Christian and Baecher, 1999). Christian and Baecher (1999) shows that caution must be paid when the output distribution is changed severely by the calculation process. This may be the case when both elasticity and plasticity are involved in the calculation Duncan and Sleep (2014) state that generally higher P f values are obtained for PEM than for Monte Carlo simulations. Chapter 2.3 MSc Thesis S.P. Kamp 35

44 Comparison Reliability-Based design and Eurocode 7 If reliability-based methods are compared with Eurocode 7 design, it becomes clear that there is no need to specify partial factors. Partial factors in Eurocode 7 account for the uncertainty in parameters, where this is done explicitly reliability-based design with probability distributions. However, it is possible to back-calculate partial factors. In order to accomplish this, the mean value, standard deviation, and the design point of a parameter are required. These are subsequently used to calculate a partial factor for that particular variable with Equation 2.5. The design point is obtained with the First Order Reliability Method, as explained in Appendix B. With reliabilitybased design it is possible to specify a higher target reliability index than defined in Eurocode 7 in order to obtain a safer design, if the consequences of failure are high. This is not possible in Eurocode 7 design, because partial factors itself are based on one certain target reliability index. Low (2014) states that there are several situations where reliability-based design is especially favourable over Eurocode 7. For uncertain parameters which are less common, such as the insitu coefficient (K) in underground excavations in rocks and the dip direction, there are no partial factors proposed in the Eurocode. This makes the use of reliability-based design particularly useful to model uncertainty. Generally, the partial factor on the unit weight (γ) is equal to one, implying there is no need to account for uncertainty. This parameter can be determined or estimated accurately generally. If there is uncertainty to account for in this parameter, reliability-based design can be used efficiently. Reliability-based design is also convenient when the sensitivity of input parameters is problemdependent. For example, strength parameters govern the output in slope stability problems, where the influence of stiffness parameters is negligible. Sometimes there is the need to model the correlation, positively or negatively, between parameters (e.g. cohesion and friction angle). In current Eurocode 7 designs is not possible to take correlation into account, and the same design is obtained with or without considering correlations. It is also not necessary to determine characteristic and design soil parameters with reliability-based used design, since the whole range of the parameter is taken into account with reliability-based methods. 36 MSc Thesis S.P. Kamp Chapter 2.3

45

46 2.4 Theory structures An often seen structure in geotechnical engineering is a retaining wall. In this research the focus will be partly on cantilever retaining walls. Cantilever retaining walls are popular since they take less space and are generally cheaper than gravity walls. However, cantilever walls usually need reinforcement to resist bending moments as a result of the horizontal soil pressure. The design must satisfy two requirements, namely internal and external stability must be guaranteed. Internal stability is ensured by providing sufficient resistance against bending moments and shear forces, where external stability considers geotechnical failure. Hydraulic failure modes, such as piping and heave are not considered here. The structural design with respect to reinforcement will also not be covered in this research. One research where this aspect is taken into account is done by Babu and Basha (2008). In this section relevant aspects of retaining walls, e.g. dominant soil properties, are discussed. Figure 2.21: Typical retaining wall, with sliding and virtual planes Lateral earth pressure The purpose of a retaining wall is to retain soil and to resist the lateral pressure of the soil against the wall. Most lateral pressure theories are based upon the sliding soil wedge theory. In simple terms, this theory is based upon the assumption that when the wall is suddenly removed, a triangular wedge of soil will slide down along a sliding plane. The soil wedge will induce a pressure on the wall, which will result in a displacement of the wall if the pressure is large enough. In case the wall does not displace with respect to the soil or ongoing compaction of the soil due to vibrations, the neutral earth pressure should be used. Using active soil pressure, especially in serviceability limit states, would underestimate the soil pressure. The neutral soil pressure is computed as follows: K 0 = (1 sin φ ) OCR (2.48) where OCR is the over consolidation rate. It should be noted that this formula should not be used for high values of OCR. The neutral soil pressure can also be used when the displacement of the retaining wall is insufficient, or when there is ongoing compaction of the soil behind the wall. Assuming active soil pressure instead of neutral soil pressure would be an unsafe assumption. Especially in serviceability limit states the neutral soil pressure should be used. In the case of ultimate limit states it is assumed that using active soil pressure is valid. Due to increasing soil pressure, the structure will deform, which will lead to a decrease in soil pressure. When a retaining wall is close to other structures, e.g. in urban areas, a higher neutral soil pressure can be used. The active soil pressure is assumed to act on the virtual plane on the back of the retaining wall (Figure 2.21). The active soil pressure, when straight virtual planes are assumed, is then computed as: β = Angle of slope backfill φ = Angle of internal friction K a = [ cos 2 α 1 + cos 2 (φ + α) sin(φ +δ)sin(φ β) cos(α δ)cos(α+β) ] 2 (2.49) 38 MSc Thesis S.P. Kamp Chapter 2.4

47 α = Wall slope angle from horizontal (90 for vertical face, α 90 if the wall is battered outward or 90 if battered inward) δ = Angle of friction between the soil and structure (generally assumed δ = 2/3φ If the backfill is levelled, both equations simplify to: K a = 1 sin φ 1 + sin φ (2.50) When the retaining structure is displaced as a result of the soil wedge, passive soil pressure will occur in front of the wall. In the case of cantilever retaining structures this pressure is often neglected since it is a favourable action and its magnitude is relatively small. The passive soil pressure is calculated as: K p = [ cos 2 α 1 cos 2 (φ α) If the backfill is level, the equation above simplifies to: sin(φ δ)sin(φ +β) cos(α δ)cos(α+β) ] 2 (2.51) K p = 1 + sin φ 1 sin φ (2.52) The load vector acting at the virtual plane has an angle equal to the ground surface, independent of the friction angle of the soil. Therefore, the vertical load component will be equal to zero for a horizontal ground surface. The vertical component will only be larger than zero when β = δ > Failure mechanisms There are distinguished four main failure mechanisms, namely sliding, bearing capacity failure, overturning and structural failure (Figure 2.22). Most important failure mechanisms in this research are the geotechnical failure mechanisms. When geotechnical failure mechanisms of cantilever retaining walls are considered analytically, it is general use to consider the soil above the heel as part of the construction for simplicity. This is indicated at the left side of the virtual plane. The virtual plane should not be confused with the sliding planes, which are also indicated in the figure. The soil part above the toe at the left side is generally neglected. Reasons for neglecting this passive soil pressure in the analyses are: Limited magnitude Sufficient displacement is required for passive soil pressure to develop Uncertainty construction Figure 2.22: Concept figure failure mechanisms retaining walls Chapter 2.4 MSc Thesis S.P. Kamp 39

48 Zevgolis and Bourdeau (2010) examined the stability a retaining wall considering it as a system consisting of different failure modes. The failure modes are due to overturning of the wall about the toe, due to sliding along the base, and due to inadequate bearing capacity of the foundation soil. Sliding occurs due to the horizontal earth pressure which tends to slide the retaining wall along its base. The force is resisted by the shear force created between the structure and the soil. The resisting force depends on the applied vertical load on the base and adhesion between foundation base and the soil. The failure modes will not be discussed in detail here. Evaluating the stability is complicated due to (geotechnical) actions and the number of possible combinations of these actions. Each time it should be analysed which combination is the most unfavourable. A retaining wall should be checked on the following failure mechanisms: Bearing capacity subsurface, as with a shallow foundation. against overturning. Sliding Overall stability (deep sliding planes, Bishop s method). This also provides the check Often a surcharge is present on the backfill of the retaining wall (e.g. a building, a crane or a road). The surcharge will exert an extra force on the retaining wall. As discussed in Section 2.1, it is often difficult to determine whether this surcharge has a stabilizing or destabilizing effect on the construction. This depends the magnitude and direction of the surcharge, the construction phase and the location of the surcharge with respect to the virtual plane. When a horizontal force is present, it is still straightforward to determine whether it is favourable or unfavourable. A horizontal force is always unfavourable, since it is the driving force for sliding failure. For bearing capacity failure it is also unfavourable, since the overturning moment and the inclination of the load vector are maximum, which is unfavourable for eccentricity and the depth of the failure plane. For an inclined backfill slope it becomes more complicated. Vertical soil pressure components of the backfill provide resistance against overturning on the one hand, and provide an additional vertical pressure on the other hand. The first contribution of the vertical component is therefore favourable for checking bearing capacity, the second is unfavourable. In advance it cannot be known if a minimum or maximum value of the vertical actions is governing. Besides an analysis with partial factors, an analysis without partial factors and without surcharge is required. The following partial factors for retaining walls are defined in NEN Annex A: Table 2.5: Partial factors from NEN Annex A for retaining walls. Soil parameter Symbol Set M2 Friction angle γ ϕ 1.2 Cohesion γ c 1.5 Undrained shear strength γ cu 1.5 Volume unit weight γ γ Important variables Several studies have been performed on reliability-based design of retaining walls. Several studies show that the friction angle of the backfill is the most sensitive parameter (Mandali et al. (2011), Guha Ray et al. (2014)). Babu and Basha (2008) shows that the most sensitive soil properties, with respect to the reliability index, are the friction angle and unit weight of the backfill for all failure modes, except for bearing capacity. For bearing capacity, the cohesion is a more important factor. In the same research the effect on the wall dimensions is also shown for all failure modes. For example, in case of an increasing coefficient of variation of the friction angle, the length of the heel and toe of the retaining wall should be increased to maintain to same reliability index. It is important to bear in mind that these results are obtained for that particular case (i.e. specific dimensions and soil properties) and one should always pay attention when using this results. 40 MSc Thesis S.P. Kamp Chapter 2.4

49 However, this research provides a good insight in the reliability-based design of retaining walls and the influence of changes in parameters. Chapter 2.4 MSc Thesis S.P. Kamp 41

50

51 Chapter 3 Verification Point Estimate Method 3.1 Introduction Literature study provided an efficient reliability-based method to derive the reliability of geotechnical structures, the so-called Point Estimate Method (PEM), which may be applicable in engineering practice to evaluate the reliability of geotechnical structures. The Point Estimate Method is a well-documented probabilistic method, which was originally developed by Rosenblueth (1975). The principles of this method are discussed in Section 2.3. In this research a feature from the finite element model Plaxis 2D is adopted in such a way, it can be regarded as the Point Estimate Method. Although PEM is a proven reliability-based method, applicability of this method in Plaxis must be verified. In order to be confident it delivers correct results it is checked against Monte Carlo simulations. The Parameter Variation feature in Plaxis is adopted in this research to perform a probabilistic analysis for ultimate limit state design. Utilizing probabilistic methods for reliability-based design allows engineers to compute the reliability index (β) of geotechnical structures. The applicability of PEM is verified by comparing statistical parameters, such as the mean (µ) and standard deviation (σ), from PEM against Monte Carlo simulations. In order to efficiently perform the verification process, an appropriate methodology is set up. Thereafter, the Point Estimate Method will be verified on the basis of several typical geotechnical structures. Analytical methods and finite element programs with probabilistic features will be used to verify the Point Estimate Method. Verification is performed for several benchmarks by computing statistical output parameters, based on different limit state functions. Limit state functions are typically based on the factor of safety, displacements or structural forces. Each benchmark serves a certain purpose with the view on the final case study. First a traditional geotechnical problem is analysed, namely a slope stability problem. Several variations will be considered for this problem, such as assuming uncorrelated and correlated input parameters. For all slope stability problems the reliability index is based on the factor of safety. Subsequently, verification is carried out for a shallow foundation where the reliability index is based on the bearing capacity. Final benchmark is a concrete cantilever retaining wall. The verification procedure and computed reliability index are on the basis of the factor of safety and displacements. In addition to the verification procedure, a comparison analysis is carried out between the Point Estimate Method and Eurocode 7. Goal of this comparison is to investigate the limitations and possibilities of the Point Estimate Method. Design standards such as Eurocode 7 are based on experience and previous standards. Therefore it is desirable that the use of probabilistic methods is consistent with previous practice and that it provides similar results. It is investigated how the captured behaviour with PEM compares to Eurocode 7 and how this information can be interpreted and used in geotechnical design. 3.2 Verification methodology The methodology of the Point Estimate Method is elaborated in this section, as well as the verification procedure. Elaboration of the methodology starts from the point where the problem geometry is already defined in the finite element program. It is assumed that mean values (X m ) of input parameters are known. To verify the model performs as expected, initially a deterministic calculation with mean values is performed. Logically the mean value obtained with a deterministic calculation 43

52 should be (practically) similar to the value obtained from the PEM output distribution. This deterministic calculation is also performed with verification software. Performing a calculation with mean values allows to calibrate both models for an appropriate verification. This calibration is an important element of a reliable verification procedure. When this is completed successfully, it should be determined which input variables have a major influence on the computational results. The importance of performing this analysis in advance of PEM is to reduce the number of random variables. The number of calculations increases exponentially with the number of random variables in PEM. In order to determine the significance of parameters, a Sensitivity Analysis is carried out for certain criteria. Parameters that show significant influence, i.e. variables with a sensitivity score above a certain threshold value, are assumed as random in further research. Other parameters will be kept at their deterministic value. The Sensitivity Analysis is shortly discussed below, background information can be found in Appendix D. The Sensitivity Analysis requires to define two evaluations points, a lower and a upper value. For convenience in further research, the evaluation points will be chosen to be located at one standard deviation below and above the mean value, or µ σ and µ + σ, respectively. Conveniently, these locations are also the evaluation points in the Point Estimate Method (Section 3.1). The mean value of input variables is assumed to be known initially. Typical coefficients of variation (COV ) can be obtained from literature, soil investigation or design standards. The standard deviation can then be calculated with the use of the coefficient of variation (COV = σ/µ). To compute all combinations, the Sensitivity Analysis requires 2n + 1 calculations. For each calculation one variable is varied at a time, other parameters are kept at their mean value. Input parameters that show a higher sensitivity score than a certain threshold value, should be taken as stochastic in further research. This threshold value is often a arbitrary choice. Generally a threshold value of 5% or 10% is used. Figure 3.1: Locations evaluation points Point Estimate Method. Subsequently, the Parameter Variation feature is utilized to perform the Point Estimate Method including the random variables that followed from the Sensitivity Analysis. As mentioned before, two evaluation points need to be defined prior to the Point Estimate Method. As originally defined by Rosenblueth (1975), these evaluation points are located at µ σ and µ+σ. In the case of PEM all possible combinations of the input parameters are calculated (non-random parameters are kept at their mean value). If n is the number of stochastic variables, 2 n calculations need to be executed. Reason to keep the number of random variables restricted, is that the number of calculations increases rapidly. For example, 2 random variables requires only 4 calculations, where including 5 random variables already leads to 32 calculations. After all calculations are completed the results can be analysed. For each combination the results can be evaluated in terms of a certain limit state function (e.g. the safety factor or bendings moments). The mean and standard deviation of the output are computed and a probabilistic distribution function has to be assumed. Generally, a normal or lognormal distribution is assumed in geotechnical engineering (Section 2.2). Finally, the reliability index can be computed and the reliability of the geotechnical structure is assessed. A flowchart of the verification methodology is shown in Figure 3.2 and a generalized template is shown in Appendix E. 44 MSc Thesis S.P. Kamp Chapter 3.2

53 Reliability-Based Ultimate Limit State Design in Finite Element Methods Figure 3.2: Flowchart verification methodology. Chapter 3.2 MSc Thesis S.P. Kamp 45

54 Verification software Verification is carried out on the basis of comparison with other finite element programs and proven analytical methods. Besides the finite element software Plaxis, other software is used. For the slope stability problem the software Slide6.0 (Rocscience) is used, which calculates the factor of safety of slopes on the basis of limit equilibrium theory. The program allows to use various analytical methods, such as Bishop and Fellenius. In this research Bishop s limit equilibrium method will be applied. One of the most appropriate and accurate ways to verify probabilistic methods is with Monte Carlo simulations, which will be used for this reason in this research. The Monte Carlo method provides useful information about the exact shape of output distributions, where the Point Estimate Method is incapable to provide this information. Another finite element program that will be utilized is Phase2 (Rocscience). This program will be used to verify the cantilever retaining wall in the third benchmark. This benchmark is verified on the basis of the factor of safety and displacements. The probabilistic analysis in Phase2 is executed with Monte Carlo simulations. In this verification procedure it is important both methods provide comparable results. Therefore each benchmark is calibrated such that it provides similar results for the mean values of input parameters. During the verification procedure it is important to keep in mind that errors may occur. Many sources of uncertainty are involved in finite element modelling and probabilistic methods. For example, in the case of slope stability analysis different analytical methods to assess the factor of safety can be used, e.g. Bishop or Fellenius. Since finite element methods are used, there are several other causes of discrepancy, such as the mesh size. Serious effort has been made to keep all errors as small as possible. Differences in the results may occur as a consequence of: Model uncertainty Rounding errors Type of analytical method. In Slope 6.0 the adopted analytical method is Bishop, where also other methods such as Spencer and Fellenius are possible. Number of simulations with Monte Carlo method. Uncertainty propagation in Point Estimate Method Human errors Methodology comparison with Eurocode 7 After each benchmark has been verified, the output of PEM is compared with Eurocode 7. Since the partial factor method from the Eurocode may be hard to apply in some cases, ideally the use of partial factors would be unnecessary in practice. In geotechnical engineering it is common practice that new standards are based on previous standards and experience. Desirably the proposed reliability-based method delivers consistent and similar results as the partial factor method. Therefore each benchmark is designed according to Eurocode 7 and PEM. The behaviour of the structure will be compared for both methods. The following steps will be performed in this procedure: For the comparison of PEM with Eurocode 7, the following steps will be executed: Derive characteristic soil properties (X k ) from the mean soil properties (X m ) (X k = X m 1.64 σ x ). The characteristic value is defined as the lower 5% fractile in Eurocode 7. In order to obtain this value the coefficient of variation (COV ) of the soil property is required. Derive design soil properties (X d ) from the characteristic soil properties (X k ). Design properties are obtained by factoring the characteristic properties by their partial factor (γ m ) (X d = X k /γ m ). Partial factors are defined for typical geotechnical structures in NEN Annex A (CEN, 2012). Perform deterministic calculations in Plaxis with the mean, characteristic and design value. 46 MSc Thesis S.P. Kamp Chapter 3.2

55 Next, the characteristic and design value are obtained from the probability density function computed with PEM. The characteristic value is again defined as the 5% fractile, the design value defined as the 0.1% fractile (Gulvanessian et al., 2012). Subsequently, a comparison is made between Eurocode 7 and PEM. The computed characteristic and design values are compared to analyse in what degree they are consistent with each other. This can be done for both normal and lognormal distributions of the PEM output. Figure 3.3: Schematization methodology comparison Point Estimate Method with Eurocode 7. Figure 3.3 shows the comparison procedure. The input distributions represent soil property distributions. The top left distribution shows the derivation of characteristic and design values according to Eurocode 7. Two deterministic calculations are performed with these values, which results in two deterministic results, EC7 k and EC7 d. The bottom left distribution shows the Point Estimate Method input, i.e. the evaluation points. Calculation with PEM results in an output distribution. From this distribution the characteristic (P EM k ) and design (P EM d ) value are obtained, the 5% and 0.1% fractile respectively. The values obtained with Eurocode 7 are indicated on this distribution for comparison purposes. Partial factors in Eurocode 7 are computed with the use of calibration analyses. These partial factors are calibrated in such a way they can be applied on a large range of geotechnical problems. Although PEM is compared with Eurocode 7, it is not known if Eurocode 7 is the exact solution. This is due to the fact that partial factors are based on experience and calibrated for general applications. Goal of this comparison is to see to what extend the results from EC7 and PEM agree. Obviously it is not evident that the same results are obtained. However, when a new method is introduced, it is desirable it is in line with previous standards. Assumptions and limitations Due to limited research time and clarity of this research, several simplifying assumptions are made. Most relevant assumptions are listed below: Chapter 3.2 MSc Thesis S.P. Kamp 47

56 Only soil properties are assumed random. Generally soil properties are assumed to be uncorrelated and normally distributed. Including structural properties would lead to many random variables in the analysis, and therefore long computation times. Besides the main goal is to include the uncertainty in soil properties. The geometry and soil properties are chosen such that they represent realistic problems. The sensitivity of input parameters largely depends on the defined lower and upper bound. Other parameters may prove to be important when different ranges are applied (Brinkgreve et al., 2016c) The linear elastic perfectly-plastic model with Mohr-Coulomb failure criterion is used in Plaxis. One of the main reasons is the amount of input variables. For example, the Hardening Soil model already requires multiple stiffness parameters For every benchmark it is assumed the groundwater level is located at great depth, meaning it does not influence the results. For simplicity only normal and lognormal distributions are assumed, both for input and output variables. It must be kept in mind that other results may be other when other distributions shapes are used. Although PEM is compared with Eurocode 7, it is not evident that Eurocode 7 is the correct solution. Partial factors resulted from experience and probabilistic analysis and calibrated for general applications. However, new standards are desirably in line with previous standards, which makes the comparison of PEM with Eurocode 7 logical. When PEM is compared with Eurocode 7, Design Approach 3 from Eurocode 7 is used to obtain the design values The characteristic value is assumed as the 5% fractile, the design value as the 0.1% fractile. 48 MSc Thesis S.P. Kamp Chapter 3.2

57 3.3 Benchmark 1: Slope stability problem One of the most common geotechnical problems is a slope stability problem. The principles of slope stability analysis are not discussed here, but can be found in Das (2010). This benchmark verifies whether the Point Estimate Method provides reliable results when the factor of safety is assessed by means of the ϕ c reduction in Plaxis. This method is generally used to evaluate the overall stability of slopes and other geotechnical structures. Various situations will be analysed, e.g. normal and lognormal input distributions of soil properties and uncorrelated and correlated variables. Results obtained with Plaxis are compared with Monte Carlo simulations, which are carried out in Slide6.0. The reliability index is based on the factor of safety, which is determined with Bishop s limit equilibrium method in Slide6.0. In order to evaluate the safety factor in Plaxis, an ϕ c reduction is performed Problem geometry and soil properties The basic geometry of this benchmark is shown in Figure 3.4. A slope with a height of H = 10 m and a slope angle of 2 : 1 is considered. Since the stability will be evaluated, mainly strength properties are relevant. The mean soil properties (µ X ) and coefficients of variation (COV X ) are shown in Table 3.1. The slope consists of one layer and is assumed to be uninfluenced by the groundwater level, since it is located at great depth. Both methods are calibrated to the extent that they provide similar results for mean input parameters. A factor of safety of is obtained in Plaxis with the ϕ c reduction, Slide6.0 provides a safety factor of with Bishop s limit equilibrium method. This limited deviation is assumed to be acceptable. Figure 3.4: Geometry slope stability problem. Table 3.1: Mean soil properties: normal distribution. Property Symbol Unit Mean µ X COV X Unit weight γ [kn/m 3 ] 21 - Friction angle ϕ [ ] Cohesion c [kp a] Normal input distributions All input parameters are taken to be normally distributed in the first case. As mentioned in the verification procedure, before executing PEM, it is evaluated which input parameters have a major influence on the computational results. Parameters identified with a significant influence are taken stochastic in further analysis to account for uncertainty. It is commonly known that in the case of slope stability problems, the cohesion (c ) and friction angle (ϕ ) have the largest influence on the factor of safety. However, for completeness of the procedure a Sensitivity Analysis is performed. The criterion for this Sensitivity Analysis is the factor of safety. This analysis requires to define a lower and upper bound. As with the Point Estimate Method in further research, the lower and upper boundary are taken one standard deviation below and above the mean value (µ ± σ). In Chapter 3.3 MSc Thesis S.P. Kamp 49

58 order to determine the standard deviations, the coefficients of variation (COV = σ/µ) from Table 2b from NEN are used (Appendix A). All possible combinations of stochastic parameters are computed, while varying only one parameter at a time. The results of the Sensitivity Analysis are shown in Figure 3.5. For this benchmark the threshold value of the sensitivity score is set to 10%. It can be clearly seen that the most sensitive parameters in this particular case are the cohesion c and friction angle ϕ, and in some degree the volumetric weight γ. Since the volumetric weight falls below the threshold value of 10%, only the cohesion and friction angle are assumed as stochastic parameters in further analysis Sensitivity score [%] γ ϕ c E ν Input variable Figure 3.5: Sensitivity analysis: factor of safety criterion. Next, the reliability index (β) of this slope stability problem is evaluated for a constant COV c and a range of COV ϕ values. The procedure will be elaborated in detail for a constant COV c = 0.20 and COV ϕ = 0.10 here. First the evaluation points, which are located at µ σ and µ + σ, need to be defined. Every possible combination of these evaluation points is analysed. In the case of two stochastic variables this results in 2 n = 4 combinations. For each combination a ϕ c reduction is performed in Plaxis to compute the factor of safety of that particular combination. From these four calculations the mean factor of safety (µ msf ) and the standard deviation (σ msf ) can be calculated. All evaluation point combinations and their corresponding factors of safety for this problem are shown in the top half of Table 3.2. Since input variables are uncorrelated, all output values have the same weight (P i = 0.25) (Section 2.3.3). In order to compute the reliability index, a limit state function has to be defined. More background information about limit state functions can be found in Section 2.3. When the factor of safety is evaluated, structures are considered to be safe when the factor is larger than unity. This requirement can be translated into a limit state function. With this requirement the limit state function for this problem is defined as: Z = µ msf 1 (3.1) 50 MSc Thesis S.P. Kamp Chapter 3.3

59 Table 3.2: All possible combinations of evaluation points and their corresponding factor of safety (µ ϕ = 35, COV ϕ = 0.10, µ c = 5 kp a, COV c = 0.20). Combination ϕ [ ] c [kp a] Msf [ ] µ msf σ msf β msf The same problem is investigated in Slide6.0 by means of Monte Carlo simulations. Due to this large amount of simulations it is possible to fit probability density functions to the output data. The obtained data is fitted to three distribution shapes, namely a normal, gamma and lognormal distribution (Figure 3.6). Monte Carlo simulations show that the best-fit distribution is a Gamma distribution in this case. However, it can be observed graphically that all distributions look similar, due to the low coefficient of variation (COV < 0.30) (Schneider and Schneider, 2012). Therefore, only normal and lognormal distributions will be considered for the sake of simplicity in further research. Considering the defined limit state function in Equation 3.1, little effort is required to compute the reliability index. The definition of the reliability index, i.e. the number of standard deviations from the mean value to the limit state (Z = µ msf 1), is graphically displayed in Figure 3.6. Assuming the factor of safety follows a normal distribution, the reliability index (β normal ) is calculated as: β normal = µ msf 1 σ msf (3.2) When the factor of safety is assumed to be lognormally distributed, the reliability index (β log ) is follows from: ( ) µ ln msf 1+COV 2 msf β log = (3.3) ln(1 + COVmsf 2 Probability density function (pdf) β σ msf µ msf Normal Lognormal Gamma Factor of safety Msf [-] Reliability index β [-] β normal β log Monte Carlo COV ϕ [-] Figure 3.6: Probability density functions for COV c = 0.20, COV ϕ = 0.10, µ msf = and σ msf = (left) and results of Point Estimate Method in Plaxis and Monte Carlo simulations for COV c = 0.20 and a range of COV ϕ values (right). Chapter 3.3 MSc Thesis S.P. Kamp 51

60 Figure 3.6 shows the results for a constant COV c = 0.20 and a range of COV ϕ values. This figure also shows the results obtained by performing Monte Carlo simulations in Slope 6.0. It can be seen in Figure 3.6 (right) and Table 3.3 that the reliability indices obtained with the Point Estimate Method in Plaxis are in good agreement with the Monte Carlo results. Especially when the factor of safety is taken normally distributed, a small and acceptable error is found between PEM and Monte Carlo. It can be seen in Figure 3.6 that the reliability index is overestimated when a lognormal is assumed. This simple illustrates that even when two distributions look similar on first sight there may be a significant difference. In order to prevent an overestimation of the overall safety, it is important to choose the distribution appropriately. By analysing various fitted distribution shapes a reliable estimation can be made of the actual output. Table 3.3: Error of Reliability index (β normal ) between Point Estimate Method and Monte Carlo simulations for COV c = 0.20 and a range of COV ϕ. COV ϕ P EM β normal MC β normal Error [%] Lognormal input distributions In geotechnical engineering often normally distributed soil properties are assumed for mathematical simplicity. However, in some cases it would be more appropriate to assume a lognormal distribution. For example, when negative values are mechanically impossible in reality, a lognormal distribution is chosen to prevent the occurrence of negative values. A soil property where this may be applicable is the cohesion. In this analysis a lognormal distribution is assumed for both the friction angle and cohesion. Now a lognormal distribution is assumed, the evaluations points are no longer located at µ ± σ. The distribution is no longer symmetrical, which must be accounted for in the evaluation points. In order to adjust for the shape of the distribution, weights are assigned to the standard deviations. These weights are the so-called standard deviation units (ξ xi ). The locations of evaluation points for lognormal distributions are calculated according to Equations 3.4 to 3.5. The weights of evaluation points (P i = 0.25) do not change with respect to the normal distribution, only the location of the evaluation points itself. The skewness depends on the coefficient of variation (COV ) of the input parameter distribution (Benjamin and Cornell, 2013): ν xi = 3 COV + COV 3 (3.4) The skewness is then used to calculate to locations of the evaluation points: ξ xi = ξ xi+ ν xi ξ xi+ = ν ( x i ( νxi 2 ) 2 ) 1/2 (3.5) x i = µ xi ξ xi σ xi x i+ = µ xi + ξ xi+ σ xi (3.6) The same geometry and soil properties (Table 3.1) are used as in the case of normal input distributions. The evaluation points are calculated below. Using Equation 3.4, the skewness of both parameters is: ν ϕ = 3 COV ϕ + COV 3 ϕ = = (3.7) ν c = 3 COV c + COV 3 c = = (3.8) Applying the skewness as calculated above in Equation 3.5, the following standard deviation units are obtained: ξ ϕ = ξ ϕ + = 1.16 (3.9) ξ c = ξ c + = (3.10) 52 MSc Thesis S.P. Kamp Chapter 3.3

61 These standard deviation units are then used in Equation 3.6 to compute the locations of the evaluation points for the friction angle: ϕ = µ ϕ ξ ϕ σ ϕ = = ϕ + = µ ϕ + ξ ϕ + σ ϕ = = (3.11) And the following evaluation points for the cohesion are computed: c = µ c ξ c σ c = = 4.26 kp a c + = µ c + ξ c + σ c = = 6.35 kp a (3.12) Table 3.4: Results lognormally distributed input variables: statistical parameters and comparison. µ msf σ msf β normal β log P EM M C Error [%] The results are shown in Table 3.4. When stochastic input parameters are non-normally distributed, it is more likely the output does not follow a non-normal distribution either. Monte Carlo simulations prove that the best-fit for the output is a lognormal distribution. It can be seen in the table above that both methods show practically similar results. However, as with normal input distributions, attention must be paid to the output distribution shape. Graphically there is only little difference between a normal or lognormal distribution, as can be seen in Figure 3.7. This figure also illustrates the reliability index for the normal distribution. However, numerically there can be observed a significant deviation in Table 3.4. Again the safety is overestimated for a lognormal distribution. Even though a lognormal distribution proves to be a better fit, the question is whether it is worth the additional computational effort in engineering. Applying a lognormal distribution is mathematical more complicated compared to a normal distribution. Especially for low coefficients of variation (COV < 0.30), a normal distribution proves to provide satisfying results generally. Probability density function (pdf) β σ msf µ msf Normal Lognormal Gamma Factor of safety Msf [-] Figure 3.7: Distributions fitted on output for lognormally distributed input variables. Chapter 3.3 MSc Thesis S.P. Kamp 53

62 3.3.4 Correlated cohesion and friction angle In reality certain properties are often correlated. In this variation on the original benchmark the possibility to take correlation into account in FEM with PEM is investigated. One of the most common correlations to consider in geotechnical engineering, is the correlation between the cohesion c and the friction angle ϕ. These soil properties are negatively correlated, i.e. a high friction angle is associated with a low cohesion, and vice versa. The general theory for including correlation with PEM is explained more detailed in Section One advantage of including correlation in PEM is that the input parameters do not change with respect to uncorrelated parameters. With respect to an uncorrelated PEM analysis, only the weights that are applied to the output are changed. In the case of two uncorrelated parameters, which results in four calculations, the weights are all equal to P i = In this analysis the input parameters and evaluation points are equal to those used in Section 3.3.2, i.e. normally distributed. The evaluation points are shown again in Table 3.5. The factors of safety in this table were all computed for uncorrelated variables, but can now be easily converted to correlated variables. Figure 3.8 illustrates a schematization where the relationship between correlated variables is shown. Two stochastic input parameters lead to four different parameter combinations, and therefore requires four calculations. Each calculation corresponds to a certain weight P i. For uncorrelated variables the weights are identical conveniently (P i = 1/2 n ). Weights for other correlations can be found in Table 3.6. The general formula to calculate weights for correlated variables can be seen in Equation The underlying principle of these correlation weights will now be explained for a correlation of ρ ϕ c = 0.5. Table 3.5: All possible combinations of evaluation points and their corresponding factor of safety (µ ϕ = 35, COV ϕ = 0.10, µ c = 5 kp a, COV c = 0.20). Combination ϕ [ ] c [kp a] Msf [ ] It can be seen in Figure 3.8 that higher weights (P 2 = P 3 = 0.375) are found when the location of one variable is one standard deviation below the mean, and the other above. This can be clearly observed for Combination 2 (P2) where ϕ is at (µ σ) and c at (µ + σ), and Combination 3 (P3) where ϕ is at (µ + σ) and c at (µ σ). The lower weights are obtained when both variables are located one standard deviation below or above the mean value, i.e. Combination 1 (P1) and Combination 4 (P4). When real soil behaviour is now considered, it is often seen that a low cohesion corresponds to a high friction angle, and vice versa. This relation explains the higher weights for these combinations in this example. The combinations with one lower and one upper value are more likely to occur in reality and therefore are assigned with a higher weight. Due to this higher weight, these combinations have a larger contribution in the final results. When the weights are known, the mean value and standard deviation can be computed with Equation 3.13 and Equation 3.14, respectively. µ xi = 2 n i=1 P i y i (3.13) σ xi = 2n P i (y i µ) 2 (3.14) i=1 54 MSc Thesis S.P. Kamp Chapter 3.3

63 Table 3.6: Weights for correlated ϕ and c. Figure 3.8: Location weights for correlated ϕ and c. ρ ϕ c P1 P2 P3 P As in preceding analyses, verification is done with Slide6.0 by means of Monte Carlo simulations. The results of PEM and the Monte Carlo simulations are shown in Table 3.7. It can be seen that both the mean value and standard deviation show satisfactory results. The reliability index is directly related to these parameters and therefore also shows satisfactory results. For this problem it can also be observed that a larger negative correlation leads to a larger reliability index. This is especially the result of a lower standard deviation for a increasing negative correlation, due to the weights that are assigned to the lowest and highest factor of safety. Since the factors of safety for Combination 1 and Combination 4 are cancelled (P 1 = P 4 = 0), the spread in the factor of safety decreases. Altogether it is shown with this example that correlation between input parameters can be taken into account efficiently with PEM. Table 3.7: Results from analysis with correlated input parameters: statistical parameters and comparison. P laxis Slide 6.0 Error [%] ρ ϕ c µ msf σ msf β msf Chapter 3.3 MSc Thesis S.P. Kamp 55

64 3.3.5 Layered cohesive slope The next variation considers undrained conditions for a multi-layered problem. The same geometry is used, only the lower 10 m is referred to as the foundation layer in this case (Figure 3.9). This problem shows the accuracy in the case of a layered geometry and undrained behaviour. Both layers have a thickness of 10 m and the slope angle is again 2 : 1. In this case the only stochastic variables are the undrained shear strengths c u of both layers. The foundation layer is assigned an undrained shear strength c u1 = 100 kp a and the top layer an undrained shear strength c u2 = 50 kp a. For both layers the coefficient of variation is equal to COV cu = Since there are two stochastic parameters, four combinations of input variables are possible. These PEM calculations, as well as the resulting factors of safety from these calculations are shown in Table 3.8. Figure 3.9: Geometry layered slope stability problem. The output is assumed to be normally distributed. The computed mean value (µ msf ), standard deviation (σ msf ) and reliability index (β normal ) resulting from these PEM calculations are listed in the same table. Also the mean value, standard deviation and reliability index from the Monte Carlo simulation and the error with PEM are included here. It can be observed that the results are fairly similar. The high standard deviation is a result of the large coefficient of variation of the input variables. In this case it might be better to assume a lognormal distribution for the output. It is shown for a undrained, layered slope stability problem that this method again proves to perform satisfactory. Table 3.8: Results multi-layered analysis: statistical parameters and comparison. Combination c u1 [kp a] c u2 [kp a] Msf [ ] P EM [ ] MC [ ] Error[%] µ msf σ msf β normal Comparison with Eurocode 7 One of the goals of this thesis is to develop a reliability-based design method which makes the application of partial factors unnecessary. This section evaluates how the Point Estimate Method compares to Eurocode 7. To be able to use PEM in combination with or as an alternative to Eurocode 7, both methods should be consistent in some degree. In order to compare both methods, deterministic Plaxis calculations with characteristic and design values according to Eurocode 7 are performed. It must be noted that characteristic values are only used for volumetric weight (γ), friction angle (ϕ ) and the cohesion (c ). Other parameters showed negligible influence, as followed from the sensitivity analysis, and are therefore kept at their mean value. Comparison is only done for the case discussed in Section 3.3.2, where only normal input parameters were assumed. 56 MSc Thesis S.P. Kamp Chapter 3.3

65 Table 3.9: Partial factors from NEN Annex A for slope stability problems. Slope stability set M2 Reliability Class (RC) Soil parameter Symbol RC1 (β T = 3.3) RC2 (β T = 3.8) RC3 (β T = 4.3) Friction angle a γ ϕ Cohesion γ c Undrained shear strength γ cu Volume unit weight γ γ a This factor is applied to tanϕ. Initially mean soil properties are used to determine the characteristic values with Equation The standard deviations (σ X ) are derived from the mean properties (µ X ) and coefficients of variation (COV X ). The design properties for the deterministic calculations according to Eurocode 7 are derived by factoring the characteristic properties with the partial factors, as defined in Table 3.9. For slope stability problem three Reliability Classes (RC) are distinguished. Each reliability class is assigned a different target reliability index which must be met in order to ensure stability. However, focus is not on whether the slope is safe enough, but on how PEM and Eurocode 7 compare. In the comparison with Eurocode 7, the PEM variation from Section is used. Characteristic and design values of input parameters for this case are listed in Table 3.10 for all Reliability Classes. Factors of safety obtained with deterministic calculations (column EC7) are also listed in this table. Table 3.10: Characteristic and design input soil properties and resulting safety factor for all Reliability Classes (COV ϕ = 0.10, COV c = 0.20, COV γ = 0.10). P EM ϕ [ ] c [kp a] γ [kn/m 3 ] EC7 Normal Lognormal X m X k X d RC RC RC Factors of safety obtained with the partial factor method can be compared with PEM (two most right columns). These safety factors are obtained from the probability density function which was produced with the PEM output data. Commonly used distributions are the normal and lognormal distribution. When the output is assumed to be normally distributed, the characteristic value (lower 5% fractile) is defined as: X k = µ X 1.64 σ X (3.15) The design value for a normal distribution, when assumed as the lower 0.1% fractile, is defined as: X d = µ X 3.09 σ X (3.16) When the output is assumed to be lognormally distributed, the characteristic value (5% fractile) is defined as: µ X k = ( 1.64 exp ) ln(1 + COV 2 ) (3.17) 1 + COV 2 The design value for a normal distribution, when assumed as the 0.1% fractile, is defined as: µ X d = ( 3.09 exp ) ln(1 + COV 2 ) 1 + COV 2 (3.18) Chapter 3.3 MSc Thesis S.P. Kamp 57

66 It is shown in Table 3.10 that similar results are obtained with PEM as with the partial factor method from Eurocode 7. The results are also graphically shown in Figure In this figure the distribution of the first benchmark variation (Section 3.3.2) and the results obtained by using the partial factors for RC2 are shown. As expected due to the relatively linear process, mean values are practically similar. The safety is slightly overestimated by PEM for characteristic values, meaning Eurocode 7 is more conservative in this case. Three different Reliability Classes are defined in Eurocode 7. When probabilistic methods are used the design value is just defined as the 0.1% fractile (Gulvanessian et al., 2012). This fractile is compared with the three deterministic values for different Reliability Classes. However, since these classes are based on different reliability indices, it would be logical and rational to also use different fractiles to compare design values. Since Eurocodes only define one fractile-value for the design value, the 0.1% fractile will be used here. Also the differences between both methods to account for uncertainty make it an acceptable choice. A significant difference is obtained between these three classes. The value obtained with PEM is within the range of all classes, but is closest to factor of safety obtained with RC2, which is a logical Reliability Class for this type of slope. Advantage of PEM is that it provides a specific target reliability index. With Eurocode 7 a target reliability index is reached by applying the partial factors. It is concluded that satisfying results are obtained for the design value by using the 0.1% fractile. Probability density function (pdf) P EM d EC7 k EC7 d P EM k Normal Factor of safety Msf [-] Figure 3.10: Comparison Eurocode 7 and PEM Conclusion In this first benchmark, which considers a traditional slope stability problem, the Point Estimate Method is verified with Monte Carlo simulations. Initially a single layered slope with uncorrelated and normally distributed variables is analysed. It is seen that the reliability index decreases for an increasing COV ϕ. This is a logical consequence, since a larger COV means a larger uncertainty, which in turn leads to a lower structure reliability. It can be concluded that the normal distribution provides a better fit than a lognormal distribution for the factor of safety. Assuming a lognormal distribution leads to an overestimation of the safety. Subsequently, uncorrelated and lognormal distributed input variables are assumed. When input parameters are assumed lognormally distributed the weights are unchanged, only the locations of the evaluation points are adjusted. Since non-normal distributed soil properties are assumed, it is more likely that the output is also non-normal distributed. This is due to the uncertainty propagation through the finite element model. However, for this slope stability problem it is concluded the output can be assumed to be normal distributed. It is generally agreed that for low coefficients of variation (COV < 0.30) a normal distribution is a good approximation and provides satisfactory results (Schneider and Schneider, 2012). Another variation with correlated normally distributed input variables is investigated. In this case 58 MSc Thesis S.P. Kamp Chapter 3.3

67 the cohesion and friction angle are assumed to be correlated. When variables are correlated the evaluation locations are equal to those in the uncorrelated case, but the weights are different. This convenient way of including correlation means that the results of the initial normal, uncorrelated variables can be used. Additional computational effort only concerns the application of the appropriate weights to the output, dependent on the correlation coefficient. For different correlation coefficients it is shown that PEM provides satisfactory results, compared to Monte Carlo simulations. Finally, the first case, with uncorrelated normally distributed variables (µ ϕ = 35, COV ϕ = 0.10, µ c = 5 kp a, COV c = 0.20), is compared with Eurocode 7 design. Single deterministic calculations are performed in Plaxis with characteristic (5% fractile) and design (0.1% fractile) values of soil properties. Where Eurocode 7 distinguishes different Reliability Classes with different partial factors, the whole output distribution is captured by PEM. The deterministic computed factors of safety according to Eurocode 7 are compared with the values obtained from the PEM distribution. It is observed that assuming a lognormal distribution overestimates the safety of the slope. When a normal distribution is assumed, similar results are obtained as with Eurocode 7. Eurocode 7 proves to be more conservative generally. The design value of PEM (0.1% fractile) is closest to the design value obtained with RC2, which is a logical Reliability Class for this slope. However, it may be illogical to compare one fractile with three different classes. Chapter 3.3 MSc Thesis S.P. Kamp 59

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69 3.4 Benchmark 2: Shallow foundation Another traditional geotechnical problem is a shallow foundation. The principles of shallow foundation theory are not elaborated here, but can be found in Das (2010). In the previous benchmark the reliability index was based on the factor of safety. In this benchmark the reliability index will be assessed by means of the bearing capacity. The bearing capacity is an important element in the ultimate limit state design of several geotechnical structures, such as shallow foundations, cantilever retaining walls and foundation piles. This benchmark is used to investigate the applicability of the Point Estimate Method to these types of problems. Results from the Point Estimate Method will be compared with Monte Carlo simulations of the analytical Brinch-Hansen method Problem geometry and soil properties The problem geometry and soil properties of shallow strip foundation are shown in Figure 3.11 and Table 3.11, respectively. The shallow foundation is assumed to be rough and rigid. Assuming the concrete foundation is poured directly into the ground, the soil-structure interaction is assumed to be rough. The coefficients of variation are based on the values as provided in NEN Table 2b. It is assumed that groundwater is at large depth and does not influence the shallow foundation. A surcharge of q = 10 kn/m 2 is applied on top of the soil. Since the deterministic solution of Brinch-Hansen only uses the unit weight (γ), friction angle (ϕ ) and cohesion (c ), these are all assumed stochastic. Figure 3.11: Geometry shallow foundation problem. Table 3.11: Mean soil properties: normal distribution. Property Symbol Unit Mean µ X COV X Unit weight γ [kn/m 3 ] Friction angle ϕ [ ] Cohesion c [kp a] Normal input distributions In order to verify the results provided by Plaxis it is compared with the analytical solution of the bearing capacity. The geometry of the analysed shallow foundation and mean soil properties are shown in Figure 3.11 and Table All soil properties are normally distributed. Many methods exist to analytically estimate the bearing capacity. Especially for the bearing capacity factors (N) many relationships are provided in literature. Since deviations in the results are always expected, Chapter 3.4 MSc Thesis S.P. Kamp 61

70 the standard equations from Brinch-Hansen (BH) are used. The Brinch-Hansen bearing capacity formula, assuming vertical loading, is defined as: q ult = c N c + qn q γbn γ (3.19) The bearing capacity factors are calculated with the following formulas: N c = (N q 1) cot ϕ N q = (1 + sin ϕ ) 1 sin ϕ exp(π tan ϕ ) N γ = 2 (N q 1) tan ϕ (3.20) Table 3.12: Deterministic bearing capacity for mean, characteristic and design input variables for Brinch-Hansen and Plaxis. ϕ [ ] c [kp a] γ [kn/m 3 ] N c N q N γ BH P laxis p m p k p d The problem is calibrated for the Brinch-Hansen method and Plaxis with mean values in order to obtain comparable results. An additional check is performed with the characteristic and design soil properties. The corresponding bearing capacity factors for mean, characteristic and design input soil properties are shown in Table The characteristic and design soil properties are obtained according to Eurocode 7. At this point these values are used to evaluate the applicability of Brinch- Hansen as verification. The elaboration of these values and comparison with Eurocode 7 is further discussed in Section In Figure 3.12 can be seen that the bearing capacity factors are sensitive to small deviations in the friction angle. A small deviation in the friction angle, may lead to large change of the bearing capacity factors, especially for higher friction angles. However, in Table 3.12 can be seen that Brinch-Hansen and Plaxis delivers comparable results for the characteristic and design values. Keeping the differences between both methods in mind, it is concluded that satisfactory results are obtained to utilize this method for verification. Figure 3.12: Bearing capacity factors; 1) N c, N q, N γ (NEN9997-1). In order to assess the reliability index of the shallow foundation, an appropriate limit state function must be defined. For both methods, i.e. Monte Carlo with Brinch-Hansen and PEM, the limit 62 MSc Thesis S.P. Kamp Chapter 3.4

71 state function is defined as the mean bearing capacity value minus the design bearing capacity (p d ). The considered limit state function is then defined as: Z = p p d (3.21) The following formulas apply to calculate the reliability index for a normal and lognormal distribution: β normal = µ Z σ Z β log = [ ] ln µz 1 p d 1+COVZ 2 ln(1 + COV 2 Z ) (3.22) Before PEM can be executed the evaluations points have to be defined. Since three stochastic parameters are included, eight combinations are obtained as a results (2 n calculations). The evaluation points and combinations are listed in the left half of Table For each combination an ultimate bearing capacity (q ult ) is obtained. In the right half the output statistics are shown. Both methods provide similar mean values. However, a significant difference in the standard deviation is found. The larger standard deviation for Brinch-Hansen can be related to the bearing capacity factors. As mentioned before these factors are highly non-linear. Therefore a small difference in the friction angle will lead to a large deviation in the bearing capacity factors. Since evaluation points are at µ σ and µ + σ, there is a spread of two standard deviations in the input variables. The spread in the magnitude of the bearing capacity factors is larger than the input spread, which again shows the high non-linearity. In the end this results in a large spread of the Brinch-Hansen bearing capacity. Table 3.13: All possible combinations of evaluations points and corresponding bearing capacity q ult. Combination γ [kn/m 3 ] ϕ [ ] c [kp a] q ult [kn/m 2 ] P EM MC Error [%] µ qult σ qult COV qult β normal β log To compute the reliability index one has to assume an appropriate probability density function. Monte Carlo simulations allow to make an appropriate estimation of the output probability density function. In this case only a normal and lognormal distribution are considered as possible distributions to fit. When a distribution is fitted to the data it is concluded with the so-called chi-sqaure test which distribution provides the best fit. In this case the test proves that a lognormal distribution is a better fit than a normal distribution. A relatively large coefficient of variation is obtained for the bearing capacity, which is an additional reason that a lognormal distribution provides a better fit. To gain a better understanding of the influence of the probability density function on the results, the reliability index is calculated and fitted for both distributions to show the difference (Figure 3.13). The required formulas to compute the reliability index for a normal and lognormal distribution are shown in Equation A significant error can be observed between both reliability indices in Table When the normal distribution is evaluated in Figure 3.13, it can be observed that there is a probability of negative values, which is impossible in reality. Especially for the lower tail significant deviations can be expected between both distribution. Chapter 3.4 MSc Thesis S.P. Kamp 63

72 Probability density function (pdf) β σ q µ q ,000 Bearing capacity q ult [kp a] Normal Lognormal Figure 3.13: Fitted normal and lognormal distribution on data Monte Carlo simulations (µ ϕ = 25, COV ϕ = 0.10, µ c = 25 kp a, COV c = 0.20, µ γ = 25 kn/m 3, COVγ = 0.10) Comparison with Eurocode 7 Similar to the slope stability analysis, the results are compared with the partial factor methodology from Eurocode 7. It will be investigated how the deterministic bearing capacity agrees with values obtained from the probability density function generated by PEM. The same problem definition as in the verification above is used. Partial factors for shallow foundations, taken from NEN Annex A, are shown in Table These partial factors are used to obtain the design values by factoring characteristic values. Since no distinction is made between Reliability Classes for shallow foundation, RC2 is assumed. Deterministic calculations are performed with mean, characteristic and design input parameters (Table 3.15). Table 3.14: Partial factors from NEN Annex A for shallow foundations. Soil parameter Symbol Set M 2 Friction angle a γ ϕ 1.15 Cohesion γ c 1.6 Undrained shear strength γ cu 1.35 Volume unit weight γ γ 1.0 a This factor is applied to tanϕ. All combinations of stochastic input variables and output statistics can be reviewed in Table The probability density functions generated by this analysis are used to compare PEM with Eurocode 7. The characteristic and design bearing capacity can now be easily obtained from this probability density function. The characteristic value is again assumed as the 5% fractile, the design value as the 0.1% fractile. When the output is assumed to be normally distributed, the characteristic value (5% fractile) is defined as: X k = µ X 1.64 σ X (3.23) The design value for a normal distribution, when assumed as the 0.1% fractile, is defined as: X d = µ X 3.09 σ X (3.24) When the output is taken to be distributed lognormal, the characteristic value (5% fractile) is defined as: µ X k = ( 1.64 exp ) ln(1 + COV 2 ) (3.25) 1 + COV 2 64 MSc Thesis S.P. Kamp Chapter 3.4

73 The design value for a normal distribution, when assumed as the 0.1% fractile, is defined as: µ X d = ( 3.09 exp ) ln(1 + COV 2 ) 1 + COV 2 (3.26) Table 3.15: Comparison PEM and Eurocode 7 for characteristic and design values. EC7 P EM q f ϕ [ ] c [kp a] γ [kn 3 ] P laxis q f Normal Lognormal X m X k X d In the previous section it was shown that a lognormal distribution proves to provide a better fit for the bearing capacity. The results are also shown graphically shown in Figure In this figure the values for PEM according to a normal distribution are shown. When the deterministic solutions are compared with the results in Table 3.15, it can be seen that PEM generally provides higher values. Only for the design value from the normal distribution a lower value is obtained. It can be concluded that the design bearing capacity is underestimated for a normal distribution. As mentioned before, differences must be expected for the lower tail. The error found between the design values is approximately 108%. It can be concluded from this example that one must choose the shape of the distribution appropriately. Assuming a normal distribution might be considered safe, but it would lead to uneconomic designs. Generally, assuming a normal distribution, the Eurocode 7 is more conservative for the bearing capacity Probability density function (pdf) P EM d EC7 k EC7 d Normal Lognormal P EM k ,000 Bearing capcity q ult [kp a] Figure 3.14: Comparison Eurocode 7 and PEM Conclusion In this section a shallow foundation is analysed as a benchmark. Where in the first benchmark the reliability index is assessed by mean of the factors of safety, the bearing capacity is analysed in this case. Deterministic calculations according to Brinch-Hansen are compared with Point Estimate Method results. Since the Brinch-Hansen method contains only three variables, these variables Chapter 3.4 MSc Thesis S.P. Kamp 65

74 are all assumed as stochastic. With this benchmark it is shown that the bearing capacity can be efficiently used to estimate the reliability index with PEM. The reliability index from Brinch- Hansen and the Point Estimate Method may differ slightly, which is most likely due to the high sensitivity of the deterministic bearing capacity factors. Keeping the differences between both method in mind, satisfactory results are obtained. Attention must be paid to the assumed shape of the output distribution. Assuming a normal distribution might give negative values. Since a negative bearing capacity is mechanically impossible, a lognormal distribution might be more appropriate. The bearing capacity is also calculated with characteristic and design soil properties according Eurocode 7. It is shown that PEM and Eurocode 7 provide similar results for this problem. However, assuming a normally distributed bearing capacity, the design value from PEM is underestimated compared to Eurocode 7. Considering the fact that Monte Carlo simulations proved a lognormal provides a better fit, this is a logical consequence. Even though PEM generally overestimates the bearing capacity compared to Eurocode 7, it is concluded that PEM provides useful additional information. 66 MSc Thesis S.P. Kamp Chapter 3.4

75 3.5 Benchmark 3: Cantilever retaining wall The final benchmark concerns a concrete cantilever retaining wall. The applicability of PEM is verified against Monte Carlo simulations by means of the factor of safety. With the case study in mind, where stiffness parameters are relevant, displacements statistics are verified as well in this case study. Besides geotechnical failure also structural failure may be important in the design. Structural failure is considered in this benchmark by assessing the reliability index based on the bending moments. Relevant bending moments occur at the stem of the retaining wall. Since this structure is discussed in more detail in this thesis, some general background information on retaining walls is discussed in Section 2.4. Design of the reinforced concrete wall itself is not considered in this thesis. The correct application of partial factors can be complicated when retaining walls are considered. It is often not clear whether a geotechnical action acts favourable or unfavourable, as was discussed in Section 2.1. This problem is solved with probabilistic design, since partial factors are no longer required. In this benchmark it is investigated if a reliable design can be made, based on the occurring bending moments and factor of safety. In order to accomplish this goal, limit states in terms of the factor of safety and bending moments should be defined Problem geometry and input parameters The geometry and input soil properties of the retaining wall are shown in Figure The height of the wall is H = 5 m, the width of the foundation slab is B2 = 4 m and the width of the toe is B1 = 1.5 m, the thickness of both the wall and foundation slab is d = 0.5 m. The problem consists of two soil layers, a foundation layer (10 m) and a backfill layer (5 m). The backfill layer is divided in five layers in the finite element analysis, each layer of 1 m will be added in a separate phase. It is assumed that groundwater is located at large depth and has no influence on the retaining wall. All input parameters are chosen arbitrarily, but are based on those used by Goh (1994). Mean soil properties and coefficients of variation are shown in Table Figure 3.15: Geometry cantilever retaining wall. Table 3.16: Mean material properties and coefficients of variation of foundation soil and backfill soil. Foundation Soil Backfill Soil Property Symbol Unit Magnitude Symbol Unit Magnitude COV Young s modulus E f [kp a] E b [kp a] Unit weight γ f [kn/m 3 ] 22 γ b [kn/m 3 ] Cohesion c f [kp a] 5 c b [kp a] Friction angle ϕ f [ ] 35 ϕ b [ ] Chapter 3.5 MSc Thesis S.P. Kamp 67

76 Table 3.17: Material properties reinforced concrete retaining wall. Property Symbol Unit Magnitude Normal stiffness EA [kn/m] Flexural rigidity EI [knm 2 /m] Poisson ratio ν [ ] 0.2 Thickness d [m] 0.5 Weight Foundation w f [kn/m/m] 2.5 Weight Wall w w [kn/m/m] 7.5 The reinforced concrete retaining wall is modelled as a plate element in Plaxis. The reinforced concrete is assigned a Young s Modulus (E c ) of kp a and a Poisson ratio (ν c ) of 0.2. In the finite element software, plate elements do not have any thickness. Therefore a virtual thickness (d) is automatically calculated by Plaxis (Equation 3.27). It is assumed that the retaining wall has a thickness of 0.5 m. Since the Young s Modulus is assigned already to the material, the Flexural Rigidity (EI) is chosen such that a thickness (d) of 0.5 m is obtained. d = 12 EI EA (3.27) Another point of attention when modelling concrete structures in finite element models is the weight of the structure. The weight assigned to the plate element depends on the weight of the concrete, the weight of the soil, and whether the soil is on both sides of the element. When the actual construction is surrounded with soil at both sides, e.g. the foundation slab of the retaining wall in this case, the weight is calculated as: w f = (γ concrete γ soil ) d real (3.28) When the soil is only at one side of the plate, e.g. the wall in this case, the weight is calculated differently: w w = (γ concrete 1 2 γ soil) d real (3.29) Interfaces are introduced to model the complex soil-structure interaction. In this case a concrete structure in sand is considered, resulting in R inter values ranging between 0.80 and 1.0. In this analysis R inter is kept constant at 1.0. Different construction phases are distinguished to obtain realistic stresses and deformations. Eight phases can be distinguished: Phase 0: Initial phase. In this phase the stresses in the soil are generated (K 0 -procedure). Phase 1: Construction retaining wall. For simplicity it is assumed the whole retaining wall is placed at once. Phase 2-6: Placement backfill soil. Each phase a layer of 1 m is added. Phase 7: Safety analysis. In this phase a Safety Analysis is performed to obtain the factor of safety. 68 MSc Thesis S.P. Kamp Chapter 3.5

77 3.5.2 Sensitivity Analysis Since many input parameters are involved in this problem, dominant parameters must be identified. To ensure no unnecessary parameters are included in PEM, a Sensitivity Analysis is performed to identity which parameters have a major influence on the results. Variability in structural material properties is outside the scope of this research, and therefore not included in this analysis. Furthermore it is mentioned that the Poisson ratio of both layers is kept constant at a value of ν = All other soil properties are included in the sensitivity analysis. The upper and lower bounds are defined at one standard deviation below and above the mean value. The standard deviation is derived with use of the mean value and coefficient of variation (COV X = σ X /µ X ). Table 3.18 shows the input statistics of the soil properties. This table also includes the corresponding lower and upper bounds for each property are shown. Before the Sensitivity Analysis is executed, the problem is calibrated with mean values in both programs. Calibration is required to obtain comparable results. In order to determine the sensitivity scores all possible combinations are calculated, while changing only one variable at a time. This is done for two criteria, the factor of safety and maximum displacements. One soil property may have a major influence when a certain criterion is considered, and a negligible influence at the other. For example, the Young s modulus is important when displacements are considered, but irrelevant for the factor of safety. The threshold value is set to 5%. Table 3.18: Upper and lower bounds of soil properties for sensitivity analysis. Foundation Soil Backfill Soil Property µ X COV X µ X ± σ X Property µ X COV X µ X ± σ X E f (36000; 44000) E b (18000; 22000) γ f (20.9; 23.1) γ b (19; 21) c f (4; 6) ϕ b (27; 33) ϕ f (31.5; 38.5) Criterion: factor of safety First criterion is the influence of individual parameters on the factor of safety in Phase 7. The sensitivity of soil properties on the factor of safety computed by a ϕ c reduction is analysed. The results of this analysis are shown in Figure 3.16 (left). It can be seen that most sensitive variables are the foundation soil friction angle (ϕ f ), foundation soil cohesion (c f ), foundation soil unit weight (γ f ), backfill soil unit weight (γ b ) and backfill soil friction angle (ϕ b ). From this analysis it is concluded that these variables should be assumed stochastic when the factor of safety is evaluated. However, it can be argued that the backfill soil weight can be determined accurately, and therefore be kept at the deterministic value. This issue will be addressed further on in this benchmark. Criterion: displacement The second criterion is the influence of input parameters on the displacement at a certain point. The location is chosen below the rotation point of the retaining wall, since relevant deformations are found here. The results are shown in Figure 3.16 (right). It can be seen that soil properties, which have a significant sensitivity above the threshold value, are the foundation soil Young s Modulus (E f ), foundation soil friction angle (ϕ f ), backfill soil unit weight (γ b), the backfill soil friction angle (ϕ b ) and the backfill soil Young s Modulus (E b). Again it can be questioned whether the backfill soil weight should be assumed stochastic in further analyses. Chapter 3.5 MSc Thesis S.P. Kamp 69

78 50 50 Sensitivity score [%] Sensitivity score [%] E f γ f c f ϕ f E b γ b ϕ b 0 E f γ f c f ϕ f E b γ b ϕ b Input variable Input variable Figure 3.16: Sensitivity scores retaining wall: Factor of safety criterion (left) and a displacement criterion (right) Normal input distributions Now the dominant parameters have been identified, PEM is verified against Monte Carlo simulations in Phase2. Phase2 is a finite element program with the possibility to perform probabilistic stability analyses. This feature allows to compare both methods and verify PEM for structures where soil stiffness may play an important role. Verification is first performed for the factor of safety and the corresponding reliability index. The friction angle (ϕ f ) and cohesion (c f ) of the foundation layer and the friction angle (ϕ b ) of the backfill layer are assumed as stochastic variables. Ideally both unit weights are included as well, but Phase2 does not allow to define these as random variables. Consequently, these soil properties are not included as stochastic in Plaxis in this part of the benchmark to ensure identical conditions in both models. Since three random variables are included, 16 (2 n = 8) PEM calculations are required. Both methods have been calibrated using 6-noded triangular elements, in order to achieve comparable results. Plaxis and Phase2 computed a factor of safety of and 2.10, respectively, with mean input parameters. The limit state function for the factor of safety is defined in Equation More information about the definition of the limit state function can be found in Section 2.3. Z = Msf 1 (3.30) All 8 possible combinations of random variables are shown in Table Corresponding factors of safety obtained in Plaxis and Phase2 are listed in the same table. The mean value, standard deviation and reliability index of the factor of safety are then computed from these calculations. The results from Phase2 are obtained with 500 Monte Carlo simulations, which allows to fit a distribution to the data. Convergence of the Monte Carlo simulations can be found in Appendix H. Although there are some small fluctuations, it is argued the problem is practically converged after 500 simulations. For this example it is shown that fitting a normal and lognormal distribution resuls in almost the same distribution graphically. These distributions are shown in Figure However, significant variations can be found when the reliability index is computed. The reliability index for a normally distributed factor of safety is defined as: β normal = (µ msf 1) σ msf (3.31) 70 MSc Thesis S.P. Kamp Chapter 3.5

79 Table 3.19: Stochastic parameter combinations and corresponding factor of safety and reliability index for PEM and Phase2 (500 Monte Carlo Simulations). Combination c f [kp a] ϕ f [ ] ϕ b [ ] P EM P EM P hase2 Error [%] µ msf σ msf β normal When the reliability indices from PEM and Phase2 are compared, it can be concluded that satisfactory results are obtained. The difference between the mean values of both methods is about 3%, which is found to be reasonable. The error of the standard deviation is approximately 12%, which is a significant difference. It is argued this error is acceptable, keeping the model differences in mind. Logically, an error is also found in the reliability index. However, this error is smaller than the error of the standard deviation. The decreased error for the reliability index is partly due to relative large influence of the higher standard deviation obtained with Phase2. Although the Phase2 mean safety factor is higher, this is compensated with a higher standard deviation, which in the end leads to a similar reliability index. Due the small errors in the reliability index it is concluded that the applicability of PEM is verified for this benchmark. Although there is a significant error in the standard deviation, the same behaviour is obtained. 2 Normal Lognormal Probability density function (pdf) β σ msf µ msf Factor of safety Msf [-] Figure 3.17: Fitted normal and lognormal distribution on data Monte Carlo simulations. Displacements In previous benchmarks the verification criteria were based on strength properties. In this section it is verified whether the behaviour due to the influence of the stiffness properties is captured accurately by PEM. The maximum displacements in the last construction phase are analysed. The same problem as in the previous verification is assumed, only now the stiffness parameters are introduced as stochastic. It is assumed that the stiffness of the backfill (E b ) and the foundation layer (E f ) have a coefficient of variation of COV = Again both methods have been calibrated with 6-noded triangular elements. Convergence of the Monte Carlo simulations can be found in Chapter 3.5 MSc Thesis S.P. Kamp 71

80 Appendix H. Assuming all input parameters at their deterministic mean value, Plaxis and Phase2 compute a displacement of mm and mm, respectively. Both methods have been calibrated to a reasonable extent, and the error between both methods is found to be acceptable. Most important element of the verification is the correct representation of the spread in the output by PEM. The different combinations of stochastic input parameters, corresponding displacements and statistical parameters are found in Table A significant error is found for the mean value. This error in the mean value is due to the initial calibrated mean values. A smaller error is found in the standard deviation. Keeping the differences between both methods in mind (e.g. initial stress simulation, calculations method) it is argued to be an acceptable error. The most important element of this probabilistic analysis, i.e. the output uncertainty, is captured relatively accurate. Therefore it is concluded that this verification proves that the behaviour due to uncertain stiffness properties is captured properly by PEM. Table 3.20: Comparison PEM and (500) Monte Carlo simulations: displacements. Combination E f [kp a] E b [kp a] P EM P EM Phase2 Error [%] µ [mm] σ [mm] COV [ ] Comparison with Eurocode 7 Now it is has been verified Plaxis is capable to perform a probabilistic analysis of a retaining wall, it is compared with the analytical approach in Eurocode 7. Goal of this comparison is to check whether PEM captures the same behaviour as a design according to Eurocode 7. Structural design of the structure is outside the scope of this research, only structural forces and the factor of safety are compared. Comparisons between PEM and Eurocode 7 will be made for mean, characteristic and design soil properties. The following partial factors for retaining walls are defined in NEN Annex A for retaining walls (Table 3.21). Table 3.21: Partial factors from NEN Annex A for retaining walls. Soil parameter Symbol Set M2 Friction angle a γ ϕ 1.2 Cohesion γ c 1.5 Undrained shear strength γ cu 1.5 Volume unit weight γ γ 1.0 a This factor is applied to tanϕ. As in previous benchmarks the characteristic value is defined as the 5% fractile and the design value as the 0.1% fractile. When the output is assumed to be normally distributed, the characteristic value (5% fractile) is defined as: X k = µ X 1.64 σ X (3.32) The design value for a normal distribution, when assumed as the 0.1% fractile, is defined as: X d = µ X 3.09 σ X (3.33) When the output is assumed to be lognormally distributed, the characteristic value (5% fractile) is defined as: µ X k = ( 1.64 exp ) ln(1 + COV 2 ) (3.34) 1 + COV 2 The design value for a normal distribution, when assumed as the 0.1% fractile, is defined as: µ X d = ( 3.09 exp ) ln(1 + COV 2 ) 1 + COV 2 (3.35) 72 MSc Thesis S.P. Kamp Chapter 3.5

81 Factor of safety First the factor of safety is evaluated for PEM and Eurocode 7. As in previous benchmarks, characteristic values of soil properties are defined as the 5% fractile. Design values are subsequently computed by factoring the characteristic value. The same mean properties are used as in the verification above, and are shown in Table 3.22, as well as the corresponding characteristic and design values. These values will be used to perform deterministic calculations in Plaxis. Table 3.22: Mean values, characteristic values and design values of soil properties. Foundation Soil Backfill Soil Property X m X k X d Property X m X k X d E f E b γ f γ b c f c b ϕ f ϕ b Again the deterministic results from Plaxis will be compared with the output obtained by PEM. The used probability density function for this analysis can be reviewed in Figure According to the applied partial factors the retaining wall is designed to be conform to Reliability Class 2. Reliability Class 2 prescribes a target reliability index of 3.8. For this retaining wall a reliability index of 5.51 is obtained with PEM, which meets the requirement easily. The comparison with Eurocode 7 can be seen in Table This table shows the results for various PEM analyses. From this comparison can be concluded that is important to include all uncertain parameters, in order to prevent overestimation of the safety. For PEM5 the most conservative factor of safety is obtained. However, this value is still higher than the factor of safety obtained according to Eurocode 7. It can also be seen that including the backfill unit weight as stochastic parameter, leads to a lower factor of safety. In order to save computation time, this parameter can also be excluded as stochastic, since it can be determined accurately. The comparison is also shown graphically in Figure 3.18, where can be clearly observed that Eurocode 7 is more conservative than PEM. Table 3.23: Comparison PEM and Eurocode 7 for mean, characteristic and design factor of safety. P EM2 a P EM3 b P EM4 c P EM5 d EC7 X m X k X d a P EM2 : ϕ f, ϕ b b P EM3 : ϕ f, ϕ b, c f c P EM4 : ϕ f, ϕ b, c f, γ f d P EM5 : ϕ f, ϕ b, c f, γ f, γ b Altogether it is concluded that PEM captures the correct behaviour correctly when the factor of safety is considered. The same trend as with Eurocode 7 is seen and values in the same order of magnitude are obtained. Chapter 3.5 MSc Thesis S.P. Kamp 73

82 2.5 Normal Probability density function (pdf) P EM d P EM k EC7 d EC7 k Factor of safety Msf [-] Figure 3.18: Comparison Eurocode 7 and PEM. Bending moment stem Due to the horizontal soil pressure behind the retaining wall, bending moments are generated in the wall. The maximum bending moment is found at the bottom of the wall, the so-called stem. This bending moment can be easily calculated analytically (Equation 3.36). According to this formula the bending moments only depend on the friction angle and unit weight of the backfill soil, for a constant height. However, in finite element methods solutions are a function of many variables. The question is how analytical and probabilistic determined bending moments compare to the analytical method. M s = 0.5 K a γ b h 3 3 (3.36) Bending moments are computed with deterministic Eurocode 7 calculations and PEM. This is done for mean, characteristic and design values of soil properties. The same soil properties as in the analysis above are used, which can be reviewed in Table The equation above is not used, but it rather provides a proper idea of significant variables in the analysis. The bending moments are computed for different cases to show the importance of parameters. Table 3.24: Statistical parameters and mean (M m ), characteristic (M k ) and design (M d ) bending moments for different PEM analyses. P EM2 a P EM3 b P EM4 c P EM5 d EC7 µ M σ M COV M M m M k M d a P EM2 : ϕ f, ϕ b b P EM3 : ϕ f, ϕ b, c f c P EM4 : ϕ f, ϕ b, c f, γ f d P EM5 : ϕ f, ϕ b, c f, γ f, γ b 74 MSc Thesis S.P. Kamp Chapter 3.5

83 The results of different PEM analyses are shown in Table In Figure 3.19 the results from PEM4 and the deterministic results according Eurocode 7 are plotted. First the statistical parameters of the various PEM analyses are compared. It becomes clear that the standard deviation is influenced largely by certain parameters. The difference between PEM2 and PEM3 shows that the cohesion of the foundation has a significant influence on the standard deviation. The influence of this soil property does not become clear form the analytical definition. However, when the unit weight of the foundation soil (γ f ) is included, the standard deviation decreases. Generally there is accounted for a low uncertainty in unit weights. This analysis shows that including this parameter has a positive influence on the reliability. One may expect that introducing more input uncertainty in the analysis leads to a higher output uncertainty, the opposite is true in this case. It becomes clear that it is important to adequately choose the input parameters. It must be noted that results may be different when another input distribution is used. When backfill volumetric weight (γ b ) is included, the standard deviation increases. This is an expected consequence with the analytical definition in mind. However, it may be questionable if there is any uncertainty in this property, since it is relatively simple to determine in reality. Unnecessarily including this parameter would lead to a conservative design. Figure 3.19 graphically shows the comparison of PEM4 with Eurocode 7. When the mean, characteristic and design value of the bending moment are compared with the deterministic calculations according to Eurocode 7, the same trend is encountered. As with PEM, the characteristic bending moment according to Eurocode 7 is higher than the mean value. The higher design bending moment is the results of a partial factor equal to unity for the unit weight and the influence of the friction angle. This can be clearly seen when the analytical solution is analysed (Equation 3.36). This equation includes the active soil pressure, which only depends on the friction angle. The lower design friction angle and a equal unit weight, compared to the characteristic value, deliver a higher bending moment in the end. In reality, however, the active soil pressure it not a function of the friction angle alone. Interesting is that PEM is more conservative in this case than Eurocode 7. In the comparison with the factor of safety it was the other way around. From these analyses it can be concluded that no general conclusion can be made which method is the most conservative Normal Probability density function (pdf) EC7 k EC7 d P EM k P EM d Bending moment [knm] Figure 3.19: Comparison Eurocode 7 and PEM Conclusion A typical cantilever retaining wall is analysed with the Point Estimate Method in order to asses the reliability index. Verification of performing a probabilistic analysis in Plaxis is done by means Chapter 3.5 MSc Thesis S.P. Kamp 75

84 of Monte Carlo simulations in P hase2. After verification a comparison is made with the partial factor method from Eurocode 7. Comparison of PEM and Monte Carlo simulations show that both method deliver similar results. The reliability index was is computed with use of the factor of safety. Where in Phase2 500 simulations were required for the solution to converge, PEM only required 8 calculations (since 3 stochastic parameters were considered). The mean value, which followed from the 8 separate factors of safety, proved to be practically equal, just as the standard deviation. As a consequence of the small errors in the mean and standard deviation there was a logical small error in the reliability index. It follows from distribution fitting that there is only a small negligible difference between a normal or lognormal distribution, due to the low coefficient of variation. For simplicity a normal distribution is assumed in this case. Furthermore a comparison was made between Eurocode 7 and PEM. Deterministic calculations were performed with characteristic and design soil properties and besides the characteristic (5% fractile) and design value (0.1% fractile) were obtained from the output distribution of PEM. When a lognormal distribution is assumed, the safety is slightly overestimated, compared to Eurocode 7 and a normal distribution. Therefore it is advised to choose the more conservative distribution, i.e. the normal distributed factor of safety. Assuming a normal distribution delivers similar factors of safety as Eurocode 7, although Eurocode 7 is more conservative in this case. Finally, the bending moments at the stem of the wall were compared. This is done the same deterministic calculations and PEM output distribution. It is shown that PEM shows a relatively large spread, compared to the values obtained with Eurocode 7. This finite element analysis shows that the bending moment do not only depend on the friction angle and unit weight, but also other soil properties. This shows that a proper sensitivity analysis is important. No general conclusion can be made on which method is more conservative. Where Eurocode 7 was more conservative for the factor of safety criterion, PEM is more conservative for the bending moment criterion. Although this conclusion cannot be made, valuable information is gathered Eurocode 7 calculations alongside PEM. 76 MSc Thesis S.P. Kamp Chapter 3.5

85 3.6 Conclusion Verification procedure A verification procedure is carried out to confirm the applicability of the Point Estimate Method in Plaxis. With the Point Estimate Method it is possible to perform a reliability-based analysis in Plaxis. Three benchmarks with typical geotechnical structures are evaluated. In these benchmarks the reliability index (β) is computed with the Point Estimate Method, and compared with Monte Carlo simulations. Monte Carlo is considered the best probabilistic method for verification, since no assumptions and simplifications are made. In the first benchmark a traditional slope stability problem is considered. Several variations of the initial problem were analysed in this benchmark. Variables that govern the stability of the slope are mainly the friction angle and the cohesion. Therefore, the initially assumed stochastic variables are the friction angle and cohesion. Both properties are assumed to be normally distributed. The reliability index of the slope is computed based on the factor of safety. It is concluded that Monte Carlo and PEM show similar results, which proves that PEM performs well for this type of problem. The advantage of Monte Carlo is that the output distribution is known, in contrast to PEM where there is no information about the shape of the distribution. However, due to the general low coefficient of variation, a normal distribution often proved satisfactory. Also variations, such as lognormal input variables and correlated variables, prove to provide satisfactory results with PEM. Subsequently, the reliability index was assessed for a shallow foundation on the basis of the bearing capacity. In this benchmark PEM was compared with Monte Carlo simulations of the analytical Brinch-Hansen method. Stochastic variables in this problem are all soil properties in Brinch- Hansen formula, which are the unit weight, friction angle and cohesion. This benchmark shows the importance of knowledge about the shape of the output distribution. Assuming the wrong probability density function may lead to significant under- or overprediction of the results. Monte Carlo simulations show that the best-fit of the output is a lognormal distribution. When a normal distribution is assumed, bearing capacity of the soil is significantly underestimated in lower tail of the distribution, which will result in a conservative design. Finally, a reinforced concrete cantilever retaining wall is evaluated. As in the first benchmark the reliability index is computed on the basis of the factor of safety. Due to limitations of the verification software only the friction angle and cohesion of the foundation layer and the friction angle of the backfill layer are assumed stochastic. In this analysis the fit on the Monte Carlo data of a normal and lognormal distribution show almost the same distribution. When both distributions are practically the same, the normal distribution may be chosen for simplicity. There is only a small discrepancy found between the statistical parameters of PEM and Monte Carlo. The error in the reliability index is only approximately 5%, which is considered acceptable. Besides the verification of the Point Estimate Method, comparison between PEM and Eurocode 7 is made in this chapter. Goal of this comparison is to evaluate the possibilities and limitations of PEM. Eurocode 7 uses a partial factor based method in order to calculate the design values. The advantage of PEM is that partial factors are no longer necessary. The question is how reliable the results from the assumed probability density function are compared to Eurocode 7. The uncertainty in the input parameters propagates through the finite element model, but it is unknown how. Supposing all random input variables are assumed normally distributed, it may be possible that the output distribution is also a normal distribution. However, due to non-linearity in finite element models it is also likely that it is not normally distributed, but lognormal for example. From the comparison with Eurocode 7 it can be concluded that conducting a PEM analysis alone is not sufficient. Due to the uncertainty in the output distribution, it may be possible to overestimate the safety of a geotechnical structure. Even though uncertainty exists in both input and output, PEM provides valuable information about the reliability index and probability of failure, with little additional effort. When three variables are assumed to be random, eight calculations are required with PEM. Where a design according to Eurocode 7 provides two single deterministic values (e.g. factor of safety), a whole output distribution can be generated with PEM. For the presented benchmarks it has been shown that the limited number of calculations for PEM performs satisfactory compared with Monte Carlo simulations. Chapter 3.6 MSc Thesis S.P. Kamp 77

86

87 Chapter 4 Case study: Cantilever retaining wall on piles The Point Estimate Method in Plaxis was successfully verified with three benchmarks in the previous chapter. Conclusions made in the benchmarks are utilized in this case study. This case study considers a more realistic structure than the benchmarks. This complex structure involves a combination of two failure mechanisms, which significantly complicates the design. Goal of this case study is to investigate the capabilities of the Point Estimate Method for geotechnical structures where several failure mechanisms, due to both strength and stiffness properties, are involved. 4.1 Problem description In this case study a cantilever retaining wall on piles is investigated. Soil-structure interaction plays a major role in the design of this geotechnical structure. Both strength and stiffness properties affect the behaviour of the structure. Bending moments in the piles occur as an effect of two components. Strength properties, e.g. unit weight and friction angle, induce a horizontal pressure on the retaining wall. Due to this pressure the wall rotates, and bending moments occur in the piles as a result. Secondly, a lateral pressure acts on the piles due to soil deformations. Deviations of the soil stiffness in which the foundation piles are located, will influence the structural forces in the pile. The correct application of partial factors is difficult and several design combinations must be investigated according to Eurocode 7. The question is whether the design process can be made more practical by using probabilistic methods. The behaviour as captured by PEM is compared with deterministic Eurocode 7 calculations Problem geometry and soil properties The geometry of the multi-layered problem and cantilever retaining wall on piles is shown in Figure 4.1. The soil properties can be found in Table 4.1. Since the retaining wall is founded on a clay layer, foundation piles are required. The cantilever retaining wall has similar dimensions and material properties as used in the last benchmark (Section 3.5). The length of the foundation slab is B = 4 m, the wall height H = 5 m. Both the wall and the foundation slab have a thickness of d = 0.5 m. The piles are connected to the retaining wall at ground level and the tip is located in the sand layer at a depth of D = 15 m. The backfill layer has a thickness of 5 m, the clay layer has a thickness of 10 m and the foundation sand layer is 10 m thick. The backfill layer is constructed in five phases in the finite element model. In each phase a soil layer of one meter thick is applied. The groundwater level is located at ground level. The geometry and properties of the piles can be found in Table 4.2. In this case study the Mohr-Coulomb soil model is used. All soil properties are normally distributed and assumed to be uncorrelated. The interface reduction factors (R inter ) are assumed constant throughout the analysis and therefore no COV is assigned. The piles are numbered from left to right. 79

88 Figure 4.1: Geometry case study: cantilever retaining wall on foundation piles. Table 4.1: Mean soil properties: uncorrelated, normally distributed. Property Symbol Unit Clay layer Sand layer Backfill sand (subscript c) (subscript s) (subscript b) COV [ ] Unit weight γ [kn/m 3 ] Young s Modulus E [kp a] Friction angle ϕ [ ] Cohesion c [kp a] Reduction factor R inter [ ] Table 4.2: Material properties embedded beam row and geometry foundation piles. Pile properties Pile geometry Identification Unit Top pile Pile tip E kn/m Pile 1 x γ kn/m 3 25 y Pile type massive circular pile Pile 2 x D m 0.54 y L spacing m 2.5 Pile 3 x Skin resistance Linear y T top,max kn/m 10 T bot,max kn/m 100 F max kn 2000 Value ISF Default 80 MSc Thesis S.P. Kamp Chapter 4.1

89 4.2 Sensitivity analysis As described above this case study considers a multi-layered, hence many input variables are involved. In order to gain better insight into the relevant variables, a Sensitivity Analysis is performed in this section. Attention must be paid to the analysed criterion, since some dominant variables may be overlooked when the wrong criterion is set. For overall stability the criterion is set to the influence of individual parameters on the factor of safety. The factor of safety is mainly a function of strength parameters. To obtain a better understanding of dominant parameters affecting structural forces, a stress-strain or displacement criterion is required. Bending moments in the piles occur due to horizontal soil pressures and soil deformations. Furthermore bending moments may occur due to the rotation of the cantilever retaining wall, which is induced by the horizontal soil pressures of the backfill and displacements of the soil. In order to determine which variables should be included in the probabilistic analysis, several criteria will be evaluated on relevant locations in the geometry, The locations where the sensitivity of soil properties is analysed, are shown in Figure 4.2. It must be noted that the coordinates are not exactly the same for each criteria. Displacement criteria are performed at Nodes, where stress-strain criteria are performed at Stress-Points. For the sake of consistency, these points are chosen as close as possible to each other, and noted as the same coordinates for clarity. The most relevant analyses are elaborated below. More sensitivity analyses can be found in Appendix J. Figure 4.2: Locations evaluation points Sensitivity Analysis. Criterion: Factor of safety It is generally known that the factor of safety is mainly governed by strength properties. The factor of safety is obtained by performing a ϕ c reduction. From this analysis a failure surface is obtained. The typical failure surface goes through the backfill, the clay layer and also the foundation sand layer (Figure 4.3). This particular failure surface is obtained with the mean values of all soil properties. One can obtain a first indication of the relevant parameters by analysing the layers through which the failure mechanism passes. However, it is not exactly clear in what degree the factor of safety is influenced by individual soil properties from this observation. As can be seen in this figure, only a small part of the failure surface goes through the sand layer. Therefore one may conclude that soil properties from this layer do not have a significant influence on the overall stability. Performing a Sensitivity Analysis may provide a better insight in what degree soil properties affect the stability. Chapter 4.2 MSc Thesis S.P. Kamp 81

90 Figure 4.3: Typical failure surface. Stiffness properties are excluded from the sensitivity analysis because it is known their influence can be neglected. Results of a sensitivity analysis with the stability criterion are shown in Figure 4.4. Parameters that show a sensitivity score above the pre-defined threshold value of 5% are assumed stochastic in further analysis. In this case the following soil properties will be assumed stochastic: ϕ s, γ b,unsat, γ c,sat, c c and ϕ c. As discussed before, it is possible that one may neglect the strength properties of the sand layer by evaluating the failure surface graphically. However, it is shown here that the soil properties of this layer show significant influence. To be more specific, the friction angle ϕ s even shows a value above the threshold value, meaning it should be included as stochastic in further analysis. Since γ b,unsat can be determined relatively accurately, including it as stochastic is questionable. Whether this parameter should be included as stochastic or not will be discussed further later Sensitivity score [%] γ sat,s ϕ s γ unsat,b ϕ b γ sat,c c c ϕ c Input variable Figure 4.4: Sensitivity criterion: factor of safety. Criterion: Displacements Part of this case study is to investigate the possibility to design the bending moments in the structure by means of probabilistic analysis. The sensitivity of the displacement at the top of Pile 3 is investigated to gain a better inside into the influence of parameters on the bending moments in 82 MSc Thesis S.P. Kamp Chapter 4.2

91 the pile. It is known in advance that additional soil properties, besides strength properties, affect the bending moments in the foundation pile. Figure 4.5: Typical displacement Pile 3 for mean values of soil properties. The displacement criterion is set in the final construction phase (Phase 6). The sensitivity of soil properties to the total displacement u is analysed at Point C. The results of the sensitivity analysis with the criterion set to the total displacement ( u) are shown in Figure 4.6. As can be seen here, the soil stiffness E c has a major influence on the total displacement at Point C. Also the stiffness of the lower sand layer E s has a large influence. Other soil properties with a significant influence are the unit weight of the backfill (γ b ), the cohesion (c c) and the clay friction angle (ϕ c). These five soil properties must be taken into account according to the displacement criterion. Just as the analysis above, including (γ b ) as stochastic is questionable Sensitivity score [%] γ sat,s E s ϕ sγ unsat,b E b ϕ b γ sat,c E c c c ϕ c Input variable Figure 4.6: Sensitivity criterion: displacement u at Point C. Parametric study influence soil properties on bending moments Additionally, a parametric study has been performed. This analysis is done to gain a better insight into the influence of various soil properties on the bending moments in the pile. Two deterministic calculations are performed for each parameter, a lower bound and upper bound calculation, while other parameters are kept at their mean value. Only the bending moments in Pile 3 are considered, Chapter 4.2 MSc Thesis S.P. Kamp 83

92 since the maximum bending moments are found here. Figure 4.7 illustrates a typical bending moment distribution for Pile 3. This particular bending moment distribution is obtained with the mean soil properties. The top of the pile is connected to the foundation slab of the retaining wall. A relatively small bending moment is found here, therefore only the bending moments in the pile itself are analysed. Besides, the connection can be reinforced such that this point is not governing in the design. Figure 4.7: Geometry case study: cantilever retaining wall on piles. All variations are shown in Table 4.3. This table shows the maximum bending moments in Pile 3 in the final construction stage in the clay (M c ) and sand layer (M s ). In order to appropriately compare all variations a calculation with mean values (X m ) is performed. This calculation results in M c = knm and M s = knm. From Table 4.3 can be concluded that γ s, ϕ s, E b, ϕ b, γ c and R inter,c have negligible influence on the bending moments. Soil properties that show a significant influence are E s, γ b, E c, c c and ϕ c. As discussed before it might not be realistic to include γ b in the probabilistic analyses. It can also be argued that the uncertainty in γ b will be small and can be disregarded, since one can determine this unit weight relatively accurate. In cases where soil-structure interaction is involved, several combinations of soil the stiffness should be checked. From the parametric study can be observed that a combination of a high E s and a low E c lead to the highest bending moments in the pile. This combination can be compared with a beam which is fixed at one side. This governing combination will be used in deterministic Eurocode 7 calculations. Exact values of the soil stiffness in Eurocode 7 calculations will be discussed later. Table 4.3: Parametric study on the influence of parameters on the bending moments in Pile 3. γ s [kn/m 3 ] M c [knm] M s [knm] γ c [kn/m 3 ] M c [knm] M s [knm] E s [kp a] E c [kp a] ϕ s [ ] c c [kp a] γ b [kn/m 3 ] ϕ c [ ] E b [kp a] R inter,c [ ] ϕ b [ ] MSc Thesis S.P. Kamp Chapter 4.2

93 4.3 Point Estimate Method In this section the reliability index (β x ) will be computed for different limit state functions. From the Sensitivity Analyses it became clear which soil properties need to be included to account for uncertainty of soil properties. Obviously, overall stability and bending moments in the pile are a function of various soil properties. Including wrong stochastic variables in the probabilistic analysis would not only lead to long computation times, but may also results in unrealistic results. Results and conclusions from the benchmarks will be taken into account in this case study. When doubt arises about the correct stochastic variables or the number of required stochastic parameters to obtain correct results, several PEM will be executed to compare the results. Various cases with different number of stochastic variables will be compared in order to analyse how the results are affected. The number of required calculations is equal to 2 n, where n is the number of stochastic variables included in PEM. After all calculations are completed, the mean (µ x ) and standard deviation (σ x ) (or variation σ 2 ) can be calculated with the formulas shown in Equation 4.1. In this equation, y i is the answer of calculation i, and P i is the weight that is assigned to each calculation. All weights have the same value and are equal to P = 1/2 n, since this case study considers all parameters to be uncorrelated and normally distributed. Various probability density functions can be assumed for the output statistics. In this case study only normal and lognormal distributions will be assumed. µ x = 2 n i=1 P i y i σ 2 x = 2 n i=1 P i (y i µ) 2 (4.1) β x = Z x σ x (4.2) First the reliability index is computed on the basis of the factor of safety. Thereafter the reliability index is computed with PEM for the bending moment criteria. For each calculation the maximum bending moments in the clay and sand layer are analysed. In the end the output will be fitted for different probability density functions. All stochastic assumed soil properties, with their lower and upper bounds (X and X + respectively), are shown in Table 4.4. As shown above in Section 4.2 the interface parameters R inter will be kept at a constant value. It is known that lower values of R inter result in higher bending moments (Schweckendiek, 2006). However, the comparisons in this case study are made with the same program. Therefore, it is found acceptable to neglect the uncertainty in R inter. More detailed output information can be found in Appendix L and K. Table 4.4: Stochastic soil properties and lower and upper bounds (evaluation points) in PEM. Property Unit X m COV [ ] X X + E s [kp a] ϕ s [kn/m 3 ] γ b,unsat [kp a] E c [kp a] γ c,sat [ ] c c [kp a] ϕ c [ ] Chapter 4.3 MSc Thesis S.P. Kamp 85

94 4.3.1 Factor of safety As in the previous section, the overall stability of the structure is mainly governed by strength properties. The sensitivity analysis with factor of safety criterion showed that the following soil properties significantly affect the stability and should be assumed stochastic: ϕ s, γ b,unsat, γ c,sat, c c and ϕ c. The mean values (X m ) and the lower and upper evaluation points of these soil properties, X and X + respectively, can be found in Table 4.4. When the safety is assessed by means of the factor of safety, the structure is considered safe when the factor of safety is larger than unity. In this case the limit state function is defined as: Z = µ msf 1 (4.3) The factor of safety is obtained with the so-called ϕ c reduction. For each calculation a factor of safety is computed. In order to investigate the influence of the number of stochastic parameters several cases are evaluated. The results of this comparison are shown in Table 4.5. An increasing standard deviation is observed for an increasing number of stochastic variables. The mean value is practically the same for each case (rounded to two decimals). In the case of an increasing standard deviation and a constant mean, logically the reliability index decreases (Equation 4.2). When only two variables are included, uncertainty effects may be overlooked and the reliability is overestimated. Especially between PEM3 and PEM4 a substantial increase of about 30% in the standard deviation is seen. This increase can be assigned to the fact that the unit weight of the clay (γ c ) is included as stochastic. Another increase in the output spread is obtained when γ b,unsat is assumed stochastic. As concluded from the last benchmark, it may be logical in some cases to exclude the backfill unit weight as a stochastic parameter. Excluding this variable leads to a small increase in the reliability index. It may be argued that this parameter can be determined accurately and therefore can be neglected as stochastic. Advantageous of excluding this property is the reduction of the required number of calculations by a factor 2 without significantly altering the results. In the subsequent comparison with Eurocode 7 the PEM4 results will be used. Table 4.5: Comparison various PEM cases: factor of safety criteria. P EM2 a P EM3 b P EM4 c P EM5 d µ msf σ msf COV msf β msf a PEM2: ϕ s, ϕ c b PEM3: ϕ s, c c, ϕ c c PEM4: ϕ s, γ c,sat, c c, ϕ c d PEM5: ϕ s, γ b,unsat, γ c,sat, c c, ϕ c The output data of PEM4 has been fitted to a normal and lognormal distribution in Figure 4.8. The reliability index for a normal distribution is indicated in this figure. It can be seen that for the lower tail the factor of safety is slightly overestimated by the normal distribution. Due to the low COV of the output, it is argued that assuming a normal distribution is appropriate (Schneider and Schneider, 2012). Differences are small and for the sake of simplicity a normal distribution is assumed in further research. In the next section, PEM4 is compared with the partial factor method from Eurocode 7, assuming a normal output distribution. 86 MSc Thesis S.P. Kamp Chapter 4.3

95 Probability density function (pdf) β σ msf µ msf Normal Lognormal Factor of safety Msf [-] Figure 4.8: Factor of safety output obtained with Point Estimate Method (PEM4) fitted on normal and lognormal distribution Bending moments From the sensitivity analysis and parametric study followed that structural forces are determined by both strength and stiffness properties. The reliability index (β) is now computed on the basis of bending moments in the foundation piles. This is done for the maximum bending moments in the clay (M c ) and sand layer (M s ). From previous analyses it followed that the backfill unit weight had little influence on the results. The same will be investigated for the bending moments. In this section it is investigated what effect including different variables has on the bending moments. The largest bending moments are encountered in Pile 3, hence this pile will be analysed here. Before the reliability index can be computed, a limit state function must be defined. Since the bending moments are evaluated a limiting, or maximum, bending moment must be set. This maximum bending moment depends on the pile properties. The maximum bending moment is chosen arbitrarily in this case study, since the structural design of the piles is outside the scope of this research. It must be kept in mind that the reliability is highly dependent on the defined limit state. Since the main goal of this research is to investigate the possibilities of PEM, and not computing a specific reliability index, it is considered an appropriate choice. When the reliability index is computed on the basis of bending moments, the limit state function as defined in Equation 4.4 is used. The bending moment capacity of the piles is chosen arbitrarily at M max = 180 knm. Z = M max M (4.4) The table below shows the maximum bending moments in the clay (M c ) and sand layer (M s )x. Similar trends can be seen for both bending moment statistics. It can be observed that the number of stochastic variables has a minimal influence on the mean values. More significant variations are found in the standard deviation. The standard deviation generally increases for an increasing number of stochastic variables. This is considered a logical consequence since the increasing input uncertainty propagates through the model. It can be questioned which parameters should be included as stochastic. At least both stiffness parameters (i.e. E c and E s ) with a significant influence are included. Furthermore the cohesion (c c) should be included. This parameter is generally difficult to determine accurately. It can also be observed that this parameter increases the standard deviation, as well as the mean value. As mentioned before including γ b is arguable. In Figure 4.6 can be seen that including this property affects the statistics significantly. However, Chapter 4.3 MSc Thesis S.P. Kamp 87

96 in further research it will not be included. The uncertainty in this variable is considered negligible and therefore PEM4 will be used in further research. A normal and lognormal distribution are fitted on the PEM4 data (Figure 4.9). Since the statistics are similar, only the (absolute) maximum bending moments of the pile in the clay layer are plotted. For all cases a relatively low coefficient of variation is found, which encourages the choice of assuming a normal distribution. In this figure the reliability index based on the bending moments is indicated for a normal distribution. This is the number of standard deviation to limit state Z, which is defined above. Table 4.6: Statistical parameters for bending moment criteria for various PEM cases. Bending moment M c [knm] P EM2 a P EM3 b P EM4 c P EM5 d µ Mc σ Mc COV Mc β Mc Bending moment M s [knm] P EM2 a P EM3 b P EM4 c P EM5 d µ Ms σ Ms COV Ms β Ms a PEM2: E s, E c b PEM3: E s, E c, ϕ c c PEM4: E s, E c, c c, ϕ c d PEM5: E s, E c, c c, ϕ c, γ b Probability density function (pdf) µ Mc β σ Mc Normal Lognormal Bending moment M c [knm] Figure 4.9: Bending moment output obtained with Point Estimate Method (PEM4) fitted normal and lognormal distributions. 88 MSc Thesis S.P. Kamp Chapter 4.3

97 4.4 Comparison with Eurocode 7 In the previous section the Point Estimate Method is used to obtain statistical parameters of the output, i.e. the factor of safety and bending moments. Also the probability density functions of the output were analysed. The governing number of stochastic variables followed from these analyses. Now the reliability index is determined for different criteria and the statistical parameters are known, PEM is compared with Eurocode 7. In this section the differences between both methods will be discussed. This comparison is particularly important since this structure is complicated to design according to Eurocode 7. Since the final goal is to design geotechnical structures without partial factors, the behaviour of the structure is compared. It must be kept in mind that Eurocode 7 itself is not the exact solution and partial factors are not calibrated for complex structures with a combination of failure mechanisms. However, the results still must be in the same order of magnitude. New design methods should be consistent with previous design standard to some extend. For the comparison of PEM with Eurocode 7, the following steps will be executed: Derive characteristic soil properties (X k ) from the mean soil properties (X m ) (X k = X m 1.64 σ x ). The characteristic value is defined as the lower 5% fractile in Eurocode 7. In order to obtain this value the coefficient of variation (COV ) of the soil property is required. Derive design soil properties (X d ) from the characteristic soil properties (X k ). Design properties are obtained by factoring the characteristic properties by their partial factor (γ m ) (X d = X k /γ m ). Partial factors are defined for typical geotechnical structures in NEN Annex A (CEN, 2012). Perform deterministic calculations in Plaxis with the mean, characteristic and design value. Next, the characteristic and design value are obtained from the probability density function computed with PEM. The characteristic value is again defined as the 5% fractile, the design value defined as the 0.1% fractile (Gulvanessian et al., 2012). Subsequently, a comparison is made between Eurocode 7 and PEM. The computed characteristic and design values are compared to analyse in what degree they are consistent with each other. This can be done for both normal and lognormal distributions of the PEM output. Figure 4.10: Procedure comparison Point Estimate Method and Eurocode 7. Chapter 4.4 MSc Thesis S.P. Kamp 89

98 Figure 4.10 shows the comparison procedure. The input distributions represent the soil property distributions. The top left distribution illustrates the derivation of characteristic and design values according to Eurocode 7. Two deterministic calculations are performed which results in two deterministic results, EC7 k and EC7 d, respectively. The bottom left distribution shows the Point Estimate Method input, i.e. the two evaluation points. Calculation with PEM results in an output distribution. From this distribution the characteristic (P EM k ) and design (P EM d ) value are obtained, the 5% and 0.1% fractile, respectively. The values obtained with Eurocode 7 are indicated on this distribution for comparison purposes. Table 4.7: Partial factors from NEN Annex A. Soil parameter Symbol γ ret Friction angle a γ ϕ 1.2 Cohesion γ c 1.5 Undrained shear strength γ cu 1.5 Volume unit weight γ γ 1.0 a This factor is applied to tanϕ. As discussed before, soil structure-interaction plays a major role in this case study. It was already elaborated what combination of stiffness properties results in the most unfavourable situation for the bending moments. Eurocode 7 states that when piles are located in soil which is subject to displacement, attention must be paid to the effect of different displacement phenomena. It is noted that generally an upper design value is used. Since Eurocode 7 provides no clear guidance on the values to be used, CUR166 will be adopted. CUR166 states that lower design value (E d,low ) can be obtained by dividing the mean value by a factor 2. In order to obtain the upper design value (E d,high ), the mean value is multiplied by a factor 1.5. No values are obtained for the lower and upper characteristic values. These values can be derived by using the magnitudes of the corresponding fractiles. The characteristic value is defined as the 5% fractile, which corresponds to a distance 1.64 σ X from the mean. The design value is defined at a distance 3.09 σ X from the mean. Further elaboration provides that the lower characteristic value (E k,low ) is obtained by dividing the mean value by The high characteristic value (E k,high ) is obtained by multiplying the mean value by The mean, characteristic and design values listed in Table 4.8. Most unfavourable bending moments are obtained for the combination of a low clay stiffness (E c ) and a high sand stiffness (E s ). In the comparison with Eurocode 7 only PEM4 cases will be used. The backfill of the unit weight will not be included as stochastic variable. The elaboration can be found in the equations below. The characteristic value (X k, 5% fractile) and design value (X d, 0.1% fractile) as defined as follows: X k = µ X 1.64σ X (4.5) X d = µ X 3.09σ X (4.6) When definitions for the design value from CUR166 and Equation 4.6 are combined, the following is obtained: X d,low = µ X /2 = σ X = µ X X d,high = µ X 1.5 = σ X = µ X (4.7) These values for the standard deviation are then implemented in Equation 4.5 to obtain the corresponding low and high characteristic value: 0.5 X k,low = µ X µ X = µ X /1.36 X k,high = µ X µ X = µ X 1.26 (4.8) 90 MSc Thesis S.P. Kamp Chapter 4.4

99 Table 4.8: Mean, characteristic and design soil properties. Clay layer Property Symbol Unit COV X m X k X d Unit weight γ c [kn/m 3 ] Young s Modulus E c [kp a] Friction angle ϕ c [ ] Cohesion c c [kp a] Sand layer Unit weight γ s [kn/m 3 ] Young s Modulus E s [kp a] Friction angle ϕ s [ ] Cohesion c s [kp a] Backfill layer Unit weight γ b [kn/m 3 ] Young s Modulus E b [kp a] Friction angle ϕ b [ ] Cohesion c b [kp a] In Section 4.3 was concluded a normal distribution is an appropriate assumption for all limit state functions. Due to the low coefficient of variation, the difference with respect to a lognormal distribution is acceptable (Schneider and Schneider, 2012). It must be kept in mind that a normal distribution overestimates for the lower tail and underestimates for the upper tail. In order to compare both methods characteristic and design values must be extracted from the PEM distribution. The characteristic value is defined as the 5% fractile (Equation 4.9) and the design value as the 0.1% fractile (Equation 4.10). These values are then compared with the results obtained with deterministic calculations. X k = µ X 1.64 σ X (4.9) The design value for a normal distribution, when assumed as the 0.1% fractile, is defined as: X d = µ X 3.09 σ X (4.10) Factor of safety A comparison is made between the obtained factors of safety by the Point Estimate Method and the partial factor method from Eurocode 7. Several cases with differently assumed stochastic parameters were investigated. It was concluded in the previous section that PEM4 (µ msf = 2.63, σ msf = 0.23) is the most representative for comparison. Deterministic calculations are performed with characteristic and design values according to Eurocode 7 (EC7). The results of PEM4 are shown in Table 4.9 and in Figure In this figure the normal distribution obtained with the statistics of PEM4 is shown. Also the values obtained with single deterministic calculations according to Eurocode 7 are shown here. The mean value obtained with PEM4 and deterministic calculations is practically equal (rounded to two decimals). When the characteristic and design values are compared, more significant deviations are found. The error between both methods is about 15%. It can be observed that PEM4 overestimates the safety compared to Eurocode 7. Significantly more conservative values are obtained for the design according to Eurocode 7. The design Eurocode value EC7 d is located far in the lower tail of the PEM distribution. One conclusion may be that the partial factor method is too conservative with respect to safety. It must be mentioned that also another fractile can be chosen for P EM d. Altogether it is concluded that PEM provides satisfactory results, since the same trend is found and PEM is in line with Eurocode 7 to some extend. Chapter 4.4 MSc Thesis S.P. Kamp 91

100 Table 4.9: Comparison PEM4 with Eurocode 7: factor of safety. PEM4 EC7 X m X k 2.25 (P EM k ) 1.99 (EC7 k ) X d 1.91 (P EM d ) 1.66 (EC7 d ) Normal Probability density function (pdf) P EM d P EM k EC7 d EC7 k Factor of safety Msf [-] Figure 4.11: Comparison Eurocode 7 and PEM4: deterministic EC7 values represented in PEM distribution Bending moments Besides the factor of safety, the bending moments in Pile 3 will be compared. When Eurocode 7 is adopted several calculations must be performed in order to determine the most unfavourable situation. This is especially required for the soil stiffness. In the sensitivity analysis the governing combination was found for a combination of a high E s and low E c. As discussed above the backfill unit weight (γ b ) is kept at the deterministic value. This leads to a PEM analysis with four stochastic variables (P EM4 = E s, E c, c c, ϕ c). The results are shown in Table A small difference is found in the mean values of PEM and deterministic mean calculations according to Eurocode 7. This error is in the order of 5%, which is acceptable. In Eurocode 7 calculations only one value is used, where PEM uses two values for each stiffness property. Due to the non-linearity there may be some spread in the mean value. It was shown that the highest bending moments are obtained for a high sand stiffness and a low clay stiffness. Mechanically this can be compared to a beam which is fixed at the bottom, i.e. the sand layer. The upper and lower characteristic and design values for the stiffness properties are listed in Table 4.8. Table 4.10: Comparison different partial factor cases with Eurocode 7: bending moments (µ Mc = , σ Mc = 13.58; µ Ms = , σ Ms = 15.24). M c [knm] M s [knm] PEM4 EC7 PEM4 EC7 X m X k X d MSc Thesis S.P. Kamp Chapter 4.4

101 The characteristic and design values are shown in Table It is assumed the unit weight γ b can be determined relatively accurately and is not taken as stochastic. The locations of Eurocode 7 values are plotted on the PEM4 distribution in Figure This is only done for the bending moments in the clay layer, since the bending moments in the sand layer shopw practically the same trend. It can be seen that Eurocode 7 and PEM show practically equal results. However, a significant difference is found for the design values. The error between both methods is still about only 10% for the design values. With the assumptions made in the design, such as the lower and upper bound values for the stiffness properties, it is concluded that PEM provides satisfactory results. The fact that both methods deliver similar results, indicates that the behaviour of the structure is captured well by PEM. Advantageous of PEM is that it is not required to identify the worst-case combination of stiffness parameters prior to the analysis. This comparison gives reason to believe that this combination is identified automatically with PEM. This automatic identification allows for a more objective design, although engineering judgement is crucial. However, as mentioned before, it is not known exactly whether Eurocode 7 is correct. This case study may indicate that Eurocode 7 overestimates the bending moments and is too conservative by providing a certain worst-case scenario for the stiffness properties. Altogether it can be concluded that PEM takes the uncertainty into account fairly well. The question is rather which method is more correct, which cannot be said at this point. Until further research is done, it is advised to use both PEM and Eurocode 7. This allows to obtain the most valuable information for these types of complex structures Normal Probability density function (pdf) P EM k P EM d EC7 k EC7 d Bending moment M c [knm] Figure 4.12: Comparison Eurocode 7 and PEM4: deterministic EC7 values represented in PEM distribution. Chapter 4.4 MSc Thesis S.P. Kamp 93

102 4.5 Conclusion Case Study In this case study a structure has been analysed where soil-structure interaction plays an important role. Due to uncertainty in the soil stiffness and high influence on structural forces, the soil stiffness is assumed stochastic for relevant criteria. When the limit state function is defined in terms of the factor of safety the stiffness parameters can be excluded from the probabilistic analysis. It is shown that attention must be paid to the number of stochastic variables. All uncertain variables must be included, otherwise the reliability index will be highly overestimated. When it is compared to Eurocode 7, it is shown PEM shows similar results. However, when the factor of safety is evaluated it is generally overestimated by PEM and Eurocode 7 is more conservative.. Furthermore, the reliability index is based on the bending moments in the foundation pile (Pile 3). Again the distributions can be approximated with a normal distribution, since the coefficient of variation is relatively low. It is concluded that there is a minimum of variables to be assumed stochastic. Otherwise the reliability may be significantly overestimated. It is important to exclude unnecessary parameters to prevent extreme computation times. For the characteristic and design values it is shown that the most unfavourable situation is reached for a high E s and low E c. Stiffness values are based on guidance provided by CUR166. When PEM and Eurocode 7 are compared it is concluded that similar trends and results are obtained. Characteristic values are practically similar. For design values higher values are obtained with Eurocode 7, but these are only about 10% higher. Keeping in mind that there are some subjective variables, such as the values of stiffness parameters and the design fractile, PEM provides satisfactory results. The deviation can also be the results of the fact that partial factors are calibrated to the occurrence of one single failure mechanisms. Partial factor are not applicable on this case study, since the structure concerns a complex combination of failure mechanisms. This case study provides reason to believe that the effects of soil-structure interaction are captured by the Point Estimate Method. When both methods are compared it can be concluded that PEM provides the worst-case scenario automatically. As mentioned before, the values are not exactly equal, but rather the same order of magnitude. This scenario is obtained without applying partial factors or performing different combinations. Only little additional effort is required to determine the input statistics and which input parameters are relevant to assume stochastic. Although, there is a significant difference in the design values, it is still useful to perform deterministic Eurocode 7 calculations. Combining information of both methods allows for a more rational design. Remark: Conclusions for the case study are valid for this geometry and input parameters only. Generalization of these conclusions must be handled carefully. 94 MSc Thesis S.P. Kamp Chapter 4.5

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104 Chapter 5 Conclusions and recommendations In this thesis the Point Estimate Method is verified and used for ultimate limit state design in the finite element program Plaxis 2D. Conclusions and recommendations based on the research done in this thesis are listed below. More specific conclusions can be found in the relevant chapters. Remark: Although the method is thoroughly checked for basic design examples, the author emphasizes that conclusions from the results may only be true for geometries, soil properties and statistical parameters applied in this thesis. This must be kept in mind when generalizing these conclusions for other applications. Conclusions This section summarizes the conclusions made in this thesis. Implementing reliability-based design in ultimate limit state verification proves to present many advantages over traditional partial factors methods. Valuable information about the possible behaviour of the structure is gathered without significant additional computational effort. The Point Estimate Method essentially only requires the mean and standard deviation of input parameters. Partial factors, which might be inconvenient to use in complex design situations, are removed from the design procedure. Furthermore, the limit equilibrium of Eurocode 7 does not provide a good solution for complex structures due to the combination of failure mechanisms. However, deterministic calculations according to Eurocode 7 complementary to the Point Estimate Method prove to be convenient. These calculations act as an additional control of the output distribution shape obtained with the Point Estimate Method. Since the Point Estimate method is unable to provide any information about the output distribution shape, information from deterministic calculations can be efficiently adopted to check for unrealistic results. Altogether PEM provides a straightforward and economic method to obtain failure probabilities for complex design situations. Several geotechnical structures have been analysed, for which the Point Estimate Method is verified against Monte Carlo simulations. Especially for relatively less complicated problems, such as slope stability problems, the Point Estimate Method delivers practically similar results as the Monte Carlo method. When non-linearity increases in the finite element models, PEM results differ more from Monte Carlo results. Problems where stiffness properties and soil-structure interactions play a role are less accurate. This is especially the case for the tail of output distributions, which may be captured inaccurately by PEM. However, it is still concluded that PEM provides satisfactory results for all analysed benchmarks and the case study. A proper indication of dominant uncertain properties is required prior to the Point Estimate Method. The output uncertainty can rapidly increase by including stochastic parameters. However, it is crucial to only assume parameters as stochastic when they are uncertain and dominant in the design. An indication of dominant parameters can be obtained by performing a Sensitivity Analysis. This analysis indicates which parameters have a major influence on the results, and which are irrelevant for the output and therefore can be kept at their mean value in further analysis. In order to prevent unnecessary computations, engineering judgement is essential with this analysis (e.g. unit weight of the backfill soil can 96

105 be determined relatively accurate). When uncertain variables are excluded, the spread in the output is generally reduced. Therefore, one must pay attention when several parameters with a small contribution are excluded, since the output standard deviation may be underestimated Lind (1983). However, due the to rapidly increasing number of calculations (2 n ), a trade-off must be made between accuracy and computational effort. Attention must be paid to the probabilistic density function of output data. When the factor of safety is analysed, the assumption of a normal distribution is generally acceptable but may be conservative. A lognormal distribution will generally lead to an overestimation of the safety. It is advised to fit at least a normal and lognormal distribution on the output to evaluate the differences. In most cases relatively low output coefficients of variation COV < 0.30 are obtained. In geotechnical engineering it is generally agreed that a normal distribution provides a acceptable representation of the behaviour for low COV (Schneider and Schneider, 2012). Besides basing conclusions only on observations of the distribution, one must critically analyse the output distributions and base conclusions on literature, experience and engineering judgement. It has been demonstrated that correlation can be included with the Point Estimate Method in an efficient way, which was also shown by Christian and Baecher (1999). The possibility of including correlated parameters has been shown for a slope stability problem. In this case the friction angle ϕ and cohesion c were assumed correlated. Advantage of this method is that no additional calculations are required with respect to uncorrelated variables, since the results from uncorrelated variables can be used. Only the weights for individual calculation combinations are different. In engineering practice soil properties are often assumed uncorrelated, while they may be correlated in reality. The Point Estimate Method provides similar results as the deterministic calculations according to Eurocode 7 (Design Approach 3). This is especially shown when strength properties are dominant and the reliability index is based on the factor of safety. Whether PEM underor overestimates the results compared to Eurocode 7 depends on the assumed output distribution. It is generally found that a normal distribution underestimates the results for the lower tail. Also for other limit state functions, such as bending moments in a foundation pile, comparable results are obtained. There is reason to believe that in the case study, where soil-structure interaction plays an important role, the similar correct governing behaviour is captured as with Eurocode 7. However, Eurocode 7 requires to check different design combinations to find the governing situation. Automatic identification of the governing behaviour by PEM is advantageous, since it reduces the subjectivity of designs. It must be mentioned that further research is required for a generalized conclusion. Valuable information is obtained about the possible spread of output parameters with only little additional computational effort. Although the information provided by PEM is highly valuable, it is not recommended to use this method on its own for the design of geotechnical structures at this point. Since the output distribution has to be assumed, additional uncertainty is introduced in the output. In order to perform an additional control, deterministic calculations according to Eurocode 7 are still required. By performing these calculations with mean, characteristic and design values the output of PEM can be checked. By comparing the output of the deterministic calculations with the mean, characteristic value (5% fractile) and design value (0.1% fractile) from PEM, it can be checked if realistic behaviour is captured. This is especially recommended for problems with non-linear behaviour in finite element models, since this may alter the shape of the output distribution. Recommendations Recommendations for further research are listed below. In this thesis research a reliability-based method is proposed to perform a more advanced geotechnical design. Although the Point Estimate Method has been verified and proven to be efficient for typical geotechnical structures, it is unknown how it exactly compares with Eurocode 7. One of the problems with this comparison is that Eurocode 7 itself is Chapter 5.0 MSc Thesis S.P. Kamp 97

106 no exact solution. Partial factors mainly have been calibrated on the basis of experience, but in such a way that they are applicable on many geotechnical applications. Additional research is required in order to achieve a more generalized conclusion on the comparison. Therefore, a large amount of additional case studies are required to gain a better insight in the correctness of the partial factors of the Eurocodes. The advised approach to obtain reliable results is by comparing Monte Carlo simulations with Eurocode 7 calculations. Various geotechnical structures, with a range of input parameters and corresponding statistics, should be investigated. Due to the minimal amount of approximations and simplifications this method provides the most reliable comparison. The most useful feature of Monte Carlo is that the entire output distribution is generated. More insight can be gained in whether Eurocode 7 provides rather conservative or overestimated designs. After Eurocode 7 is compared with Monte Carlo, both methods should be compared with the Point Estimate Method. Conveniently the exact output distribution is already provided by Monte Carlo, where the Point Estimate Method is unable to provide this information. With the Point Estimate Method there is significant uncertainty in the output, especially in the tails of the distribution curve. Knowing the exact shape of the output distribution allows to evaluate to what extend the correct behaviour is captured by the Point Estimate Method. At this point it advised to use the Point Estimate Method in combination with deterministic Eurocode 7 calculations. This allows to check whether PEM results are realistic or not. Results from reliability-based methods should be interpreted with knowledge from experience and engineering judgement. Above recommendation shortly describes the required comparison procedure for Monte Carlo, PEM and Eurocode 7. From this comparison it is obtained to what extent all methods agree. Conclusions with respect to the output distribution shapes of certain limit state functions of Monte Carlo should be analysed to set up guidance in assuming the output distribution with PEM. The degree of non-linearity in finite element models affects how the input uncertainty propagates through the model and how the output distribution is altered with respect to the input distribution. Although it is generally accepted that normal distributions are used for low coefficients of variations (COV < 0.30), it is desirable that general recommendations are provided for choosing the output distributions for various limit state functions, e.g. based on the factor of safety, displacements or bending moments. This allows to make a prediction for the output and if extreme outlying are to be expected prior to the analysis. In this thesis the Mohr-Coulomb soil model is used. For engineering purposes it is common to use more advanced models, such as the Hardening Soil model. Before adopting more advanced models it should be investigated how the increase in input parameters can be handled. For example, the Hardening Soil model includes several stiffness parameters, which may increase the number of stochastic input parameters, and the number of required calculations as a result. One of the advantages of the Point Estimate Method is the relatively low computational effort compared to other reliability-based methods. Another question is how the increase in model complexity is captured by PEM. Additional non-linearity will be introduced, which may increase the output uncertainty. The current version of Eurocode 7 does not provide proper guidance on the use of finite element methods for geotechnical designs. General disagreement is found on how and when to apply partial factors in the calculation process. It has to be mentioned that some guidance on ultimate limit state design in finite element models exists in the form of research papers. For example Bauduin et al. (2000), Bauduin et al. (2005) and Schweiger (2005) dedicated research to this topic. Also CUR166 contains some guidance on the use of finite element models for the design of sheet pile walls. A lack of guidance is especially found when reliability-based methods are used in combination with finite element methods. Although target reliability indices and typical property statistics (e.g. coefficient of variation) are found in Eurocode 7, there is no information on how to approach a geotechnical reliability-based design. However, EN1990 provides the basics of (structural) reliability analysis. Generally a limited amount of soil investigation is available, which complicates the proper determination of input parameters. Besides the fact that reliability-based design is not common practice and engineers are still unfamiliar with the principles, some form of guidance is desired. Transition from a partial factor methodology to a reliability-based methodology would be an opportunity to harmonize design standards, since more objective designs can be obtained. 98 MSc Thesis S.P. Kamp Chapter 5.0

107 The sensitivity analyses in Plaxis proved to be an effective method to identify important uncertain input parameters. However, only the absolute sensitivity score of analysed parameters is provided. Although this already allows for an efficient procedure to determine which uncertain parameters significantly influence the results, and therefore should be assumed stochastic in further analysis, it would be useful to see whether a parameter has a negative or positive influence on the results. The current implemented sensitivity analysis is unable to state whether a parameter works favourable or unfavourable for a certain criterion, e.g. the stability. Acquiring this information would provide valuable insight in the influence of individual parameters on the behaviour of the structure. This insight allows to make more rational designs. An example of a method that provides negative and positive influence factors is the First Order Reliability Method (FORM). More advanced sensitivity analyses are listed in Schweckendiek (2006). Chapter 5.0 MSc Thesis S.P. Kamp 99

108

109 Bibliography CUR 166 Damwandconstructies, Technical Recommendation, L.E.E.S. Andrew. Using Numerical Analysis with Geotechnical Design Codes. Modern Geotechnical Design Codes of Practice: Implementation, Application and Development, 1(1): , G.L.S. Babu and B.M. Basha. Optimum Design of Cantilever Retaining Walls Using Target Reliability Approach. International journal of geomechanics, 8(4): , G. B. Baecher and J.T. Christian. Reliability and statistics in geotechnical engineering. John Wiley & Sons, New York, C. Bauduin, B. Simpson, and M. De Vos. Some considerations on the use of Finite Element Methods in Ultimate Limit State Design. In Int Workshop on Limit State Design, LSD 2000, Melbourne, C. Bauduin, K.J. Bakker, and R. Frank. Use of finite element methods in geotechnical ultimate limit state design. In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON SOIL MECHANICS AND GEOTECHNICAL ENGINEERING, volume 16, pages , J.R. Benjamin and C.A. Cornell. Probability, statistics, and decision for civil engineers. Courier Corporation, A. Bond. Implementation and evolution of Eurocode 7. Modern Geotechnical Design Codes of Practice: Implementation, Application and Development, 1:3 14, A. Bond and A. Harris. Decoding Eurocode 7. CRC Press, ISBN R.B.J. Brinkgreve, S. Kumarswamy, and W.M. Swolfs. Plaxis Material models manual. Technical report, 2016a. R.B.J. Brinkgreve, S. Kumarswamy, and W.M. Swolfs. Plaxis Tutorial manual. Technical report, 2016b. R.B.J. Brinkgreve, S. Kumarswamy, and W.M. Swolfs. Plaxis Rereference manual. Technical report, 2016c. R.B.J. Brinkgreve, S. Kumarswamy, and W.M. Swolfs. Plaxis Scientific manual. Technical report, 2016d. CEN. NEN C1: Geotechnical design of structures Part 1: General rules, R. Chalaturnyk, P. K. Robertson, T. Elkateb, R. Chalaturnyk, and P. Robertson. An overview of soil heterogeneity: quantification and implications on geotechnical field problems. Canadian Geotechnical Journal, 40(1):15, C. Che-Hao, T. Yeou-Koung, and Y. Jinn-Chuang. Evaluation of probability point estimate methods. Applied Mathematical Modelling, 19(2):95 105, Y.M. Chew, K.S. Ng, and S.F. Ng. The effect of soil variability on the ultimate bearing capacity of shallow foundations. Journal of Engineering Science and Technology, (Special Issue on ACEE 2015 Conference August):1 13, J.T. Christian. Geotechnical engineering reliability: how well do we know what we are doing? Journal of geotechnical and geoenvironmental engineering, 130(10): , J.T. Christian and G.B. Baecher. Point-estimate method as numerical quadrature. Journal of Geotechnical and Geoenvironmental Engineering, 125(9): , J.T. Christian and G.B. Baecher. The point-estimate method with large numbers of variables. International Journal for Numerical and analytical methods in geomechanics, 26(15): , T.A. Cruse. Reliability-based mechanical design. CRC Press,

110 B.M. Das. Geotechnical Engineering Handbook. J. Ross Publishing, J.M. Duncan and M.D. Sleep. Evaluating reliability in geotechnical engineering. In Risk and Reliability in Geotechnical Engineering, page Brussels European Committee for Standardization. EN Eurocode 7 part 1, General Rules CEN/TC 250/SC7, European Committee for Standardization, Brussels A.T.C. Goh. Behavior of cantilever retaining walls. Journal of Geotechnical Engineering, 119(11): , doi: / (94) D. V. Griffiths and G.A. Fenton. Probabilistic Methods in Geotechnical Engineering. Springer Wien New York, D. V. Griffiths, Gordon A. Fenton, and Derena E. Tveten. Probabilistic geotechnical analysis: How difficult does it need to be. In Proceedings of an international conference on, probabilistics in geotechnics: technical and economic risk estimation, pages 3 20, Essen, Germany., Pub. VGE. A. Guha Ray and D. Baidya. Probabilistic Analysis of a Slope Stability Problem. In Indian Geotechnical Conference, A. Guha Ray, S. Ghosh, and D.K. Baidya. Risk factor based design of cantilever retaining walls. Geotechnical and Geological Engineering, 32(1): , ISSN doi: / s y. Haig Gulvanessian, Calgaro Jean-Armand, and Holický Milan. Designer s guide to EN 1990: eurocode: basis of structural design ISBN A Harris. Eurocode 7: Geotechnical design. In J Robers, editor, The essential guide to Eurocodes transition, pages British Standards Insitution, A.M. Hasofer and N.C. Lind. Hasofer-Lind reliability index. Journal of the Engineering Mechanics Division, 100(1): , M. Heibaum and M. Herten. Geotechnical verifications using the finite-element method? Bautechnik, 86(1):7 15, M.A. Hicks. An Explanation of characteristic values of soil properties in Eurocode 7. In Modern Geotechnical Design Codes of Practice: Implementation, Application and Development, volume 1, pages M.A. Hicks and J.D. Nuttall. Influence of soil heterogeneity on geotechnical performance and uncertainty: a stochastic view on EC7. In Proc. 10th International Probabilistic Workshop, pages , Stuttgart, M. Holický. Introduction to Probability and Statistics for Engineers. Springer Science & Business Media, ISBN Y. Honjo. Challenges in geotechnical reliability based design. In Proc. of the 3rd International Symposium on Geotechnical Safety and Risk, volume 27, page 11, Münich, Germany, ISBN M. Kayser and S. Gajan. Application of probabilistic methods to characterize soil variability and their effects on bearing capacity and settlement of shallow foundations: state of the art. International Journal of Geotechnical Engineering, 8(4): , F.H. Kulhawy. On evaluation of static soil properties. In Stability and performance of slopes and embankments II (GSP 31). In In Stability and performance of slopes and embankments II, pages ASCE, S. Lacasse and F. Nadim. Uncertainties in Characterising Soil Properties. In C.D. Shackleford, P.P. Nelson, and M.J.S. Roth, editors, Uncertainty in the Geologic Environment: From Theory to Practice, volume 1, pages 49 75, Madison, Wilsonsin, American Society of Civil Engineers. N.C. Lind. Modelling of uncertainty in discrete dynamical systems. Applied Mathematical Modelling, 7(3): , MSc Thesis S.P. Kamp Chapter 5.0

111 B.K. Low. Reliability-based design: Practical procedures, geotechnical examples and insights. In K.K. Phoon and J. Ching, editors, Risk and Reliability in Geotechnical Engineering, pages CRC Press, A.K. Mandali, M.S. Sujith, B. N. Rao, and J. Maganti. Reliability analysis of counterfort retaining walls. Electronic Journal of Structural Engineering, 11(1):42 56, ISSN S.H. Marques, A.T. Gomes, and A.A. Henriques. Reliability Assessment of Eurocode 7 Retaining Structures Design Methodology. In ISGSR 2011: 3rd International Symposium on Geotechnical Safety and Risk, pages , ISBN T.L.L. Orr. Selection of characteristic values and partial factors in geotechnical designs to Eurocode 7. Computers and Geotechnics, 26(3): , T.L.L. Orr. The story of Eurocode 7. In 14th European Conference on Soil Mechanics and Geotechnical Engineering, T.L.L. Orr. Implementing Eurocode 7 to achieve reliable geotechnical designs. Modern Geotechnical Design Codes of Practice: Implementation, Application and Development, 1:72 86, doi: / G.M. Peschl and H.F. Schweiger. Reliability Analysis in Geotechnics with Finite Elements Comparison of Probabilistic, Stochastic and Fuzzy Set Methods. In ISIPTA, volume 3, pages , K.K. Phoon. Beyond coefficient of variation for statistical characterization of geotechnical parameters. In Keynote lecture of Geotechnical and Geophysical Site Characterization, volume 4, pages , ISBN K.K. Phoon and J. Ching. Risk and reliability in geotechnical engineering. CRC Press, K.K. Phoon and F.H. Kulhawy. Characterization of geotechnical variability. Canadian Geotechnical Journal, 36(4): , E. Rosenblueth. Point estimates for probability moments. Proceedings of the National Academy of Sciences, 72(10): , C. Russelli. Probabilistic Methods applied to the Bearing Capacity Problem. PhD thesis, Universität Stuttgart, C. Russelli and P.A. Vermeer. Probabilistic methods applied to Geotechnical Engineering. In 2nd International Workshop of Young Doctors in Geomechanics, page 57, H.R. Schneider and M.A. Schneider. Dealing with uncertainties in EC7 with emphasis on determination of characteristic soil properties. IOS Press, Rotterdam, The Netherlands, ISBN T. Schweckendiek. Structural Reliability Applied To Deep Excavations: Coupling reliability methods with finite elements. PhD thesis, TU Delft, Delft University of Technology., H.F. Schweiger. Application of FEM to ULS design (Eurocodes) in surface and near surface geotechnical structures. 11th International Conference Computer Methods and Advances in Geomechanics, 4: , H.F. Schweiger. Influence of EC7 design approaches on the design of deep excavations with FEM. Geotechnik, 37(3): , H.F. Schweiger and R. Thurner. Basic Concepts and Applications of Point Estimate Methods in Geotechnical Engineering. In Springer Vienna, pages H.F. Schweiger, R. Thurner, and R. Pöttler. Reliability Analysis in Geotechnics with Deterministic Finite Elements. International Journal of Geomechanics, 1(4): , B. Simpson. Partial factors: where to apply them? In Proceedings of the LSD 2000: International Workshop on Limit State Design in Geotechnical Engineering, number November, pages 1 10, Chapter 5.0 MSc Thesis S.P. Kamp 103

112 C. Smith and M. Gilbert. Ultimate Limit State design to Eurocode 7 using numerical methods. Part I: proposed design procedure and application. Ground Engineering, 44(11):24 29, G. Tjie-Long. Common mistakes on the application of Plaxis 2D in analyzing excavation problems. International Journal of Applied Engineering Research, 9(21): , B. Valley and D. Duff. Probabilistic analyses in Phase pages 0 5, B. Valley and P.K. Kaiser. Consideration of uncertainty in modelling the behaviour of underground excavations. In 5th international seminar on deep and high stress mining, pages , S. Vesic. Bearing capacity of shallow foundations. Foundation Engineering Handbook, pages , T. Vrouwenvelder, A. Van Seters, and A. Hannink. Dutch approach to geotechnical design by Eurocode 7, based on probabilistic analyses. Modern Geotechnical Design Codes of Practice, 1: , I.E. Zevgolis and P.L. Bourdeau. Computers and Geotechnics Probabilistic analysis of retaining walls. Computers and Geotechnics, 37(3): , 2010.

113 A NEN Table 2b

114 B Random variables A simple but efficient way to model uncertainty is with the use of random variables. Instead of using deterministic input parameters in the design, random variables assume values from a probability density function (pdf). Probability distributions quantify the likelihood that a value lies in any certain interval. Commonly used distributions in geotechnical engineering are a normal and lognormal distribution. Random variables are described with statistical parameters, such as the mean and standard deviation. In geotechnical engineering many variables are based on characteristic variables of parameters. Appropriate characteristic values for these characteristics are chosen on the basis of engineering judgement, available data and type of problem to be analysed. Basic principles of characteristic values can be found in Section 2.1. Reliability analyses show that uncertainties in different soil properties all affect the reliability of a structure in a different way (Lacasse and Nadim, 1996). Some important characteristics of random variables will be discussed below. 1. Mean value. The mean value µ x, also called the first central moment, is the sum of the probability of each possible outcome of an experiment multiplied by its value. In general, if x is a continuous random variable, the mean is defined as: µ x = + xf(x)dx (B.1) where f(x) is the probability density function of x (for continuous random variables). 2. Variance. Another important characteristic of a random variable is the variance, or second central moment. The variance V ar(x) of a random variable x is the quantification of the spread around the mean value of that variable. For a continuous random variable x the variance is defined as: V ar(x) = σx 2 = (x µ x ) 2 f(x)dx (B.2) 3. Standard deviation. A more common parameter to quantify the spread is the standard deviation σ x, which is defined as the square root of the variance. This parameter proved to be an useful measure to assess the uncertainty. σ x = V ar(x) (B.3) 4. Coefficient of variation. The coefficient of variation (COV ) is a dimensionless definition of the uncertainty and is extensively used in geotechnical engineering. This non-dimensional variable describes whether the spread of a random variable is small or large relative to the mean value. The coefficient of variation is defined as the ratio of the standard deviation over the mean (Equation B.4). COV = σ x µ x (B.4) 5. Skewness. Another statistical characteristic for modelling parameters is the skewness, or third central moment. In the case of a normal (Gaussian) distribution the skewness coefficient is zero, i.e. symmetric. The skewness ν x is the degree of asymmetry of the probability density function f x (x). The skewness coefficient ν x is defined as: ν x = + (X µ x ) 3 (X µ x ) 3 f(x)dx (B.5) σ 3 x 6. Correlation. Random variables can be correlated or uncorrelated. When variables are assumed to be correlated, the likelihood of a certain value of the random variable y depends on the value of the random variable x. The covariance Cov(x, y) is comparable to the variance but measures the combined effect of how two variables vary together. The definition of the covariance is: Cov(x, y) = + + (x µ x )(y µ y )f xy (x, y)dy dx (B.6)

115 where f(x, y) is the joint probability density function of the random variables x and y. To provide a non-dimensional measure of the degree of correlation between x and y, the correlation coefficient ρ x,y is obtained by dividing the covariance by the product of the standard deviations: Cov(x, y) ρ x,y = 1 ρ x,y 1 (B.7) σ x σ y A value of 1.0 or -1.0 indicates a perfect linear correlation. Given a value of x, the value of y is known, and therefore not random. A value of zero indicates there no correlation exists between variables. A positive correlation means that when a variable increases the other variable will also increase. A negative value indicates that if one variable increases, the other variable decreases. Probability Distributions The term probability density function (pdf) refers to a function that defines a continuous random variable. Some probabilistic methods do not require to know the exact shape of the distribution, but only the first moments. In the case of Taylor s series and the Point Estimate Method only the mean and standard deviation of random variables are required. If correlation is taken into account also the correlation coefficient can be included with these methods. Although knowledge of the distribution shape is not necessarily required, it is always advantageous to gather as much knowledge about the exact distribution shape as possible. A probability density function has the property that for any x, the value of f(x) is proportional to the likelihood of x. A requirement of a probability density function is that the area below the probability density function is always equal to unity. The probability that the random variable x lies between two values x 1 and x 2 is the integral of the probability density function between those values: P r(x 1 < x < x 2 ) = x2 x 1 f x (x)dx (B.8) The cumulative distribution function (cdf), or F x (x), measures the integral of the probability density function from minus infinity to x. This means that for any value of x, F x (x) is the probability that the random variable is less than a certain x. The cumulative distribution function is defined as: F x (x) = x f x (x)dx (B.9) Normal distribution The normal (Gaussian) distribution is the most well-known probability density function. This probability density function is the most straightforward to use due to its symmetry and mathematical simplicity. The probability density function f x (x) of a normal distribution is defined in terms of the mean µ and the standard deviation σ as: f x (x) = 1 ] [ σ 2π exp (x µ)2 2σ 2, < x < + (B.10) When a normal distribution is fitted, the mean of the distribution is calculated as the expected value of the random variable. A disadvantage of normal distribution is that the limits are plus and minus infinity, which is inconvenient in engineering practice in some cases. However, values more than three or four standard deviations from the mean have a very low probability to occur. Therefore, when there is only little knowledge about the distribution, an approximated fit of a normal distribution can be made by assuming reasonable minimum and maximum at plus and minus three standard deviations from the mean. The normal distribution is commonly assumed to characterize variables where the coefficient of variation is less than about COV = 0.3. Relevant properties

116 symmetrical (ν x = 0) possibility of negative values more conservative for strength-properties than lognormal distribution. Figure B.1: Probability Density Function of standard normal distribution (Schweckendiek, 2006). Lognormal distribution One of the disadvantages of a normal distribution is the possible occurrence of negative values. This is especially disadvantageous in geotechnical engineering, since soil properties are generally positive. Assuming distributions which only produce positive values solves this problem. This is often done by assuming a lognormal distribution. When a random variable x is lognormally distributed, its natural logarithm, ln(x), is normally distributed. The lognormal distribution contains several features that make its choice more favourable in engineering practice: As x is positive for any value of ln(x), lognormally distributed random variables are all assumed positive It often provides a reasonable shape in cases where the coefficient of variation is large (COV > 0.3) It has a fatter tail than a normal distribution, which makes it more conservative for load variables. Figure B.2: Probability density function of lognormal distribution (Schweckendiek (2006)). The lognormal distribution for a random variable x is specified by its mean µ, standard deviation σ and the skewness coefficient ν X. The general formula for the probability density function is defined as: [ 1 f(x) = xσ 2π exp 1 ( )] (lnx µ) 2, 0 X + (B.11) 2 σ

117 Gamma distribution The probability density function of the gamma distribution is defined as: f(x) = (x/β)α 1 /exp( x/β) βγ(α) x 0 (B.12) where Γ(α) is the Gamma function, and the parameters α and β are both positive, i.e. α > 0 and β > 0. In this function α is known as the shape parameter, and β is referred to as the scale parameter. β has the effect of stretching or compressing the range of the distribution. The shape and scale parameter are computed with the mean and standard deviation of the normal distribution: α = µ σ 2 (B.13) β = σ2 µ (B.14) Fractiles of random variables One of the most useful features of probability theory in reliability-based engineering is a certain fractile of a probability density function. A fractile indicates a value of a random value which corresponds to a given probability that values smaller than the fractile occur. Holický (2013): For a given probability p, the p-fractile x p denotes such a value of a random variable X, for which it holds that values of the variable X smaller than or equal to x p occur with the probability p. When the probability distribution function is defined as Φ(x), the value Φ(x p ) is equal to the probability p. Therefore the fractile x p follows from: P (X x p ) = Φ(x p ) = p (B.15) The same definition counts for a standardised random variable U. When the random variable U is standardized, U is substituted for X and u p is substituted for x p in Equation B.15. Values for the fractiles u p of standardised random variables U are generally available in tables, as in Table B.1. Generally the fractile x p of the random variable X is calculated with the use of tables for u p. These tables are available for standardised random variables U for relevant distribution types. The fractile x p can be derived from the standardised random variable u p, which can also be found in available tables, with the following formula: x p = µ + u p σ = µ(1 + u p COV ) (B.16) where µ is the mean, σ the standard deviation and COV the coefficient of variation of the random variable X. More information can be found in Holický (2013).

118 Table B.1: Fractiles up of a standardised random variable having three parameter log-normal distribution. Probability p α

119 C Reliability-based methods Crude Monte Carlo method One of the most suitable probabilistic method, without approximations and simplifications, it the Monte Carlo method. For some problems it may be difficult to evaluate the behaviour of the performance function. In these cases the reliability index, or probability of failure, can be determined directly using Monte Carlo simulations. For each uncertain input property a probability density function is assumed, and a large number of simulations is performed with randomly selected values. For each set of random numbers, the problem is evaluated analytically. The frequency of each outcome can be plotted by means of a histogram. When sufficient simulations have been performed a probability density function can be fitted on the output histogram. In advance a limit state is determined, for example terms of the factor of safety. By counting the total number of simulations and the number of simulations that failed, it is possible to calculate the probability of failure (Equation C.1). The Monte Carlo method is straightforward, and the lack of approximations makes it an ideal benchmark to compare it to other reliability methods. However, it can be a computational and time intensive method, since generally a large number of simulations is required for the output to converge. Another disadvantage is that Monte Carlo is not capable to identify the contribution of individual variables on the output uncertainty, since each set of input variables is randomly generated. P f = n f (C.1) n where n is the total number of simulations and n f is the number of failures. In order to obtain sufficient accuracy the following requirement is given for the number of simulations: ( ) 1 n > (C.2) P f The number of failure is counted as follows: n n f = I(X 1, X 2,..., X n ) where I(X 1, X 2,..., X n ) is a function defined as: { 1 if g(x 1, X 2,..., X n ) 0 I(X 1, X 2,..., X n ) = 0 if g(x 1, X 2,..., X n ) 0 i=1 (C.3) (C.4) First Order Reliability Method The First Order Reliability Method (FORM) is a definition for the reliability analysis originally developed by Hasofer-Lind (Hasofer and Lind, 1974). The starting point of this method is the definition of the performance function Z(X), where X is a vector of random variables. The performance function can be linear or non-linear and is described below. Sometimes convergence problems may occur with FORM in the iteratively computation of the reliability index. Linear reliability functions If the covariance of the random variables is known, the mean and standard deviation are defined by: Z = a 1 X 1 + a 2 X a n X n + b (C.5) µ z = a 1 µ x1 + a 2 µ x a 3 µ x3 + b (C.6) n n σ z = a 1 a 2 Cov(X i, X j ) (C.7) i=1 j=1 Random variables are transformed to equivalent standard normally distributed variables with the following equation: U i = X i µ Xi σ Xi (C.8)

120 Distributions which are non-normal, first have to be transformed to normal distribution. The probability that Z < 0 can be computed using the standard normal cumulative distribution (Equation (2.14)). The reliability index β is defined as the distance from the origin to the failure space, which is described by a linear function. The point on the failure space with the closest distance to the origin is the point with the highest joint probability and therefore is by definition the design point. For linear reliability function the influence coefficients are defined as: α i = a iσ Xi σ z (C.9) The design point is then defined as X i = µ Xi α i σ Xi β (C.10) Non-linear limit state function In the case of non-linear failure surfaces, the shortest distance to the origin is not unique as in the case of a linear function. The minimum value of the distance from the failure surface should now be found: Minimize, D = u t u (C.11) Using the theory of Lagrange multipliers, the minimum distance is obtained by: ) β = n i=1 u i n i=1 ( δg δu ( δg δu ) 2 (C.12) ( ) where δg δu is the partial derivative evaluated at the design point u. The design point in the reduced coordinate is given by: U i = α i β (C.13) where The influence coefficients are defined as: ( δg α i = n δu i ) i=1 ( δg δu i ) 2 (C.14) The design point in the original coordinates is given by: X i = µ i α i σ Xi β (C.15) (a) Linear performance function (b) Non-linear performance function Figure C.1: Hasofer-Lind reliability index (Cruse, 1997).

121 D Relevant Plaxis features In this research the finite element program Plaxis 2D is used. This section summarizes some of the features that will be used extensively in this research, such as: Material Models Interface Strength Safety Analysis Sensitivity Analysis Parameter Variation Material Models There can be chosen from a large database of Material Models in Plaxis 2D. All material models can be found in Brinkgreve et al. (2016a). In this research only one material model is adopted, namely the Mohr-Coulomb model. The basics of this model are discussed below. Mohr-Coulomb model The linear elastic perfectly-plastic Mohr-Coulomb model involves five input parameters. The input parameters are E and ν for soil elasticity, ϕ and c for soil plasticity and ψ as the dilatancy angle. The Mohr-Coulomb model is the most simple soil model and can be used to represent a first approximation of the actual soil behaviour. The linear elastic part of the model (Figure D.1a) is based on the well-known Hooke s law. The plastic part of the model is based on the Mohr- Coulomb failure criterion. Although the increase of stiffness with depth can be taken into account, the Mohr-Coulomb model does neither include the stress-dependency, stress-path dependency, strain dependency of stiffness or anisotropic stiffness. In general, effective stress states at failure are well described by the Mohr-Coulomb model with strength parameters ϕ and c. (b) Mohr-Coulomb Yield Criterion in Principal Stress Space (Brinkgreve et al., (a) Linear elastic perfectly-plastic Mohr- Coulomb model (Brinkgreve et al., 2016a). 2016a). Figure D.1 Input parameters: Stiffness parameters: E and ν Strength parameters: c (cohesion) and ϕ (friction angle) Dilative behaviour: ψ