Probabilistic Design Tools
for Vertical Breakwaters
MAS3 - CT95 - 0041
FINAL REPORT
VOLUME IId
PROBABILISTIC ASPECTS
Edited by J.K. Vrijling April 1999 co-sponsored by Commission of the European UnionDirectorate General XII under
Printed at:
1. VOLUME IIa – HYDRAULIC ASPECTS
Chapter 1:ALLSOP, N.W.H. (1999): Introduction.Chapter 2.1: ALLSOP, N.W.H.; DURAND, N. (1999): Influence of steep seabed slopes on breaking waves for structure design. 28 pp.
Chapter 2.2: MCCONNELL, K.J. (1999): Derivation, validation and use of parameter map. 5 pp.
Chapter 2.3: CALABRESE, M.; VICINANZA, D. (1999): Estimation of proportion of impacts. 15 pp.
Chapter 3.1: VOORTMAN, H.G.; HEIJN, K.M. (1999): Wave transmission over vertical breakwaters. 9 pp.
Chapter 3.2:ALLSOP, N.W.H.; BESLEY, P.; FRANCO, L. (1999): Wave overtopping discharges. 8 pp.
Chapter 3.3:ALLSOP, N.W.H. (1999): Wave reflections. 13 pp.
Chapter 4.1: VOORTMAN, H.G.; VAN GELDER, P.H.A.J.M.; VRIJLING, J.K. (1999): The Goda model for pulsating wave forces. 5 pp.
Chapter 4.2: FLOHR, H.; MCCONNELL, K.J.; ALLSOP, N.W.H. (1999): Negative or suction forces on caissons: development of improved prediction methods. 17 pp., 1 Annex.
Chapter 4.3: BURCHARTH, H.F.; LUI, Z. (1999): Force reduction of short-crested non-breaking waves on caissons. 17 pp., 3 Annexes.
Chapter 4.4: VRIJLING, J.K.; VAN GELDER, P.H.A.J.M. (1999): Uncertainty analysis of non breaking waves. 12 pp.
Chapter 4.5: VAN GENT, M.R.A.; TORENBEEK, R.V.; PETIT, H.A.H. (1999): VOF model for wave interaction with vertical breakwaters. 11 pp.
Chapter 4.6: LÖFFLER, A.; KORTENHAUS, A. (1999): Non breaking waves - pressures on berms. 23 pp.
Chapter 5.1: KORTENHAUS, A.; OUMERACI, H.; ALLSOP, N.W.H.; MCCON-NELL, K.J.; VAN GELDER, P.H.A.J.M.; HEWSON, P.J. ET AL. (1999): Wave impact loads - pressures and forces. 39 pp.
Chapter 5.2: WALKDEN, M.; WOOD, D.J.; BRUCE, T.; PEREGRINE, D.H. (1999): Seaward impact forces. 25 pp.
Chapter 5.3: ALLSOP, N.W.H.; CALABRESE, M. (1999): Impact loadings on vertical walls in directional seas. 19 pp.
Chapter 5.4: VAN GELDER, P.H.A.J.M.; VRIJLING, J.K.; HEWSON, P.J. (1999): Uncertainty analysis of impact waves and scale corrections due to aeration. 12 pp.
Chapter 5.5: LÖFFLER, A.; KORTENHAUS, A.; WOOD, D.J. (1999): Wave impact loads - pressures on a berm. 22 pp.
Chapter 6.2: MARTíN, F.L.; LOSADA, M.A. (1999): Wave loads on crown walls. 36 pp. Chapter 6.3: MUTTRAY, M.; OUMERACI, H. (1999): Wave loads on caisson on high
mounds. 28 pp.
Chapter 7: CRAWFORD, A.R.; HEWSON, P.J. (1999): Field measurements and database. 4 pp.
Chapter 8.1: DE GERLONI, M.; COLOMBO, D.; BÉLORGEY, M.; BERGMANN, H.; FRANCO, L.; PASSONI, G.; ROUSSET, J.-M.; TABET-AOUL, E.H. (1999): Alter-native low reflective structures - perforated vertical walls. 41 pp.
Chapter 8.2: KORTENHAUS, A.; OUMERACI, H. (1999): Alternative low reflective structures - other type of structures. 29 pp.
2. VOLUME IIb –GEOTECHNICAL ASPECTS
Chapter 1: DE GROOT, M.B. (1999): Introduction.Chapter 2: KVALSTAD, T.J. (1999): Soil investigations and soil parameters. 20 pp.
Chapter 3: LAMBERTI, A.; MARTINELLI, L.; DE GROOT, M.B. (1999): Dynamics. 56 pp. Chapter 4: DE GROOT, M.B. (1999): Instantaneous pore pressures and uplift forces. 38 pp. Chapter 5: KVALSTAD, T.J. (1999): Degradation and residual pore pressures. 37 pp.
Chapter 6: IBSEN, L.B.; JAKOBSEN, K.P. (1999a): Limit state equations for stability and deformation. 20 pp., 3 Annexes.
Chapter 6, Annex B: IBSEN, L.B.; JAKOBSEN, K.P. (1999b): Permanent deformations due to impact loading. 9 pp.
Chapter 6, Annex A: JAKOBSEN, K.P.; SØRENSEN, J.D.; BUCHARTH, H.F.; IBSEN, L.B. (1999): Failure modes - limit state equations for stability. 26 pp.
Chapter 6, Annex C: LAMBERTI, A. (1999): Combined effect of dilatancy in rubble mound and caisson inertia. 9 pp.
Chapter 7: KVALSTAD, T.J.; DE GROOT, M.B. (1999): Uncertainties. 30 pp.
Chapter 8: GOLÜCKE, K.; PERAU, E.; RICHWIEN, W. (1999): Influence of design parameters - stability analysis on feasibility level. 31 pp.
3. VOLUME IIc –STRUCTURAL ASPECTS
Chapter 1: CROUCH, R.S. (1999): Introduction.Chapter 2: MARTINEZ, A.; KOVARIK, J.-B.; BERDIN, D. (1999): Structural design of vertical breakwaters - limitations of current practice and existing design codes. 37 pp. Chapter 3: VROUWENVELDER, A.W.C.M.; BIELECKI, M. (1999): Caisson reliability
during transport and placing. 36 pp.
Chapter 4: CROUCH, R.S. (1999a): In-service behaviour of cellular reinforced concrete caissons under severe wave impact. 39 pp.
Chapter 5: CROUCH, R.S. (1999b): Some observations on the durability and repair of concrete structures in a marine environment. 30 pp.
4. VOLUME IId –PROBABILISTIC ASPECTS
Chapter 1: VRIJLING, J.K. (1999): Introduction.Chapter 2: VRIJLING, J.K. (1999): Fault tree analysis of a vertical breakwater. 9 pp.
Chapter 3: SØRENSEN, J.D.; BURCHARTH, H.F. (1999): Limit state equations including uncertainties. 26 pp., 1 Annex.
Chapter 4.1: VOORTMAN, H.G.; KUIJPER, H.K.T.; VRIJLING, J.K. (1999): Economic optimal design of vertical breakwaters. 17 pp.
Chapter 4.2: SØRENSEN, J.D.; BURCHARTH, H.F. (1999): Partial safety factor system. 24 pp.
Chapter 5.1: LAMBERTI, A.; MARTINELLI, L.; DE GROOT, M.B.; GOLÜCKE, K.; VAN HOVEN, A.; ZWANENBURG, C. (1999): Hazard analysis of Genoa Voltri breakwater. 40 pp.
Chapter 5.2: VOORTMAN, H.G.; VRIJLING, J.K. (1999): Reliability analysis of the Easchel breakwater. 29 pp.
Chapter 5.3: SØRENSEN, J.D.; BURCHARTH, H.F. (1999): Other representative structures: Mutsu-Ogawara, Niigata East and West. 19 pp.
J.K. VRIJLING, H.G. VOORTMAN
Delft University of Technology, Hydraulic and Offshore Engineering Section, Stevinweg 1, NL-2628 CN, Delft, The Netherlands
E-mail: J.K.Vrijling@ct.tudelft.nl
This volume is part of the final report of the MAST III project PROVERBS, PRObabilistic design tools for VERtical BreakwaterS (February 1996 – January 1999) under contract no. MAS3-CT95-0041. The various parts of the final report are as follows (this volume in bold letters):
Volume I
OUMERACI, H.; KORTENHAUS, A.; ALLSOP, N.W.H.; DE GROOT, M.B.; CROUCH, R.; VRIJLING, J.K.; VOORTMAN, H.G (1999): Probabilistic design tools for vertical breakwaters. Balkema, Rotterdam, ca 350 pp.
Volume IIa
ALLSOP, N.W.H. (ed) (1999): Probabilistic design tools for vertical breakwaters – Hydrodynamic aspects. MAST III – PROVERBS – project. Technische Universität Braunschweig, Braunschweig, Germany, 400 pp. Volume IIb
DE GROOT, M.B. (ed) (1999): Probabilistic design tools for vertical breakwaters –Geotechnical aspects. MAST III – PROVERBS – project. Technische Universität Braunschweig, Braunschweig, Germany, 250 pp. Volume IIc
CROUCH, R. (ed) (1999): Probabilistic design tools for vertical breakwaters – Structural aspects. MAST III – PROVERBS –project. Techni-sche Universität Braunschweig, Braunschweig, Germany, 140 pp.
Volume IId
VRIJLING, J.K.(ed) (1999): Probabilistic design tools for vertical breakwaters – Probabilistic aspects. MAST III – PROVERBS – project. Technische Universität Braunschweig, Braunschweig, Germany, 170 pp.
Probabilistic design tools provide a structured framework of dealing with all relevant physical processes and uncertainties associated with coastal structures. Within the PROVERBS project, the last of the four tasks in the project was devoted to the integration of the research results from all other tasks into a consistent probability-based analysis and design framework. The main issues addressed in this volume are:
identification of all relevant failure modes and development of fault trees; identification of all relevant parameters and the description of uncertainties;
formulation of limit state equations for all failure modes on different levels of sophistication;
development of a design philosophy for the definition of acceptable failure probabilities including the development of a cost optimisation procedure;
development of a Partial Safety Factor System (PSFS) for vertical breakwaters; application of design tools to representative structures.
The results of the research of Task 4 are summarised in chapter 5 of volume I. In this volume, a more detailed background is given on all the subjects covered.
Chapter 2 of this volume covers the identification of failure modes associated with vertical breakwaters and the development of fault trees for this type of structures. No definition of failure modes is possible without defining the main function of the breakwater first. In this project, the breakwaters considered were mostly devoted to the protection of harbour basins. Therefore, the main function is defined as:
"Protection of the harbour basin against unacceptable wave action"
This is the starting point of the definition of failure modes. Basically, the failure modes can be divided in two groups:
Ultimate Limit State failure. This concerns failure modes where (a part of) the breakwater looses its structural integrity. This generally leads to a breach, which causes wave energy to enter the harbour. Generally, these failure modes are characterised by the fact that they are not easily reversed and that they lead to a large damage. Therefore, accepted failure probabilities for this type are low;
Serviceability Limit State failure. These are failure modes where the breakwater remains stable, but temporarily allows too much wave energy to enter the harbour. The consequences of this type of failure are easily reversed.
The fault trees described in Chapter 2 provide an overview of the interaction between the different failure modes. By means of the failure probability for each individual failure mode and the fault tree, it is possible to calculate an estimate of the overall failure probability.
PROVERBS but including the uncertainties which generally originate from two main sources: inherent uncertainty of the input parameters;
model uncertainty.
The inherent uncertainty is generally site specific and can be described by means of distribution functions based on measurements. Since any model represents a simplification, generally some scatter is found when the model is compared to (laboratory) measurements. This model uncertainty can be accounted for in the probabilistic analysis. Engineering estimates of the model uncertainties associated with the limit state equations are discussed in this chapter.
Chapter 4 considers two types of design procedures which are generally dependent on the availability of funding for analysis and design and thus on the size of the project. In large projects, there will be room for detailed analysis and design optimisation whereas for smaller projects one has to rely on less complicated procedures. PROVERBS has therefore developed probabilistic tools suited for both types of projects. The full probabilistic approach consists of an optimisation procedure which provides the optimal geometry and the corresponding optimal failure probability as a function of the boundary conditions and the costs of construction and the consequences of failure (Chapter 4.1).
A partial safety factor system (PSFS) has been developed for the cases where no detailed information on economic consequences is available or where research funding is scarce. The PSFS provides four safety classes and the corresponding safety factors. The partial safety factors are calibrated, such that a reasonable correspondence is found with the safety levels of existing structures (Chapter 4.2).
Chapter 5 discusses the application of the probabilistic design tools to representative structures which have been used in PROVERBS to serve as a means for co-operation between the tasks. Due to the planning within the project, the use of early versions of some models in the case studies could not be avoided. Therefore, the models presented here may differ slightly from the ones presented in Volume I and Volumes IIa through IIc. A total of three case studies is carried out. Every case study applies probabilistic tools but the emphasis of the case studies is different:
the case of Genoa Voltri concentrates on foundation modelling and dynamic behaviour of the structure (Chapter 5.1);
the Easchel case shows the influence of several model choices on the resulting failure probability (Chapter 5.2);
in the case study of the Mutsu Ogawara breakwater, the influence of subsoil properties and the availability of model tests on the resulting failure probability is shown (Chapter 5.3).
approach for vertical breakwaters. However, the variety of the developed tools and case studies provides a good guidance for the design engineer involved in this type of projects. Furthermore, the probability-based tools provide a valuable insight into the "hot spots" and the important uncertainties associated with the design, thus allowing for rational decisions on breakwater geometry and choice of safety level.
BREAKWATER
J.K.VRIJLING
Delft University of Technology,
Stevinweg 1, NL-2800 GA Delft, The Netherlands E-mail: J.K.Vrijling@ct.tudelft.nl
1. INTRODUCTION
There are several causes, which may lead to the failure of a vertical breakwater. In order to examine the interaction between the possible modes of failure and their total effect on the failure of the breakwater, a fault tree has been made. The most important failure mechanisms of vertical breakwaters and the fault tree are discussed in par. 2.
The first action of the risk analysis of a breakwater is to define its function. At an abstract level the objective of the entire risk analysis is to study the ways along which the breakwater can fail to fulfil this function. In this paper the function of the breakwater is limited to: provide tranquil water for the harbour operations of sea going vessels
In many practical cases more functions as guiding tidal currents, limiting sand transport, providing amenities etc. are attributed to the structure.
However for the purpose of this paper the top-event of the breakwater fault tree is defined as Hs-harbour> 0.20m. This may occur due to three main events:
- too much wave energy enters the harbour via the entrance by refraction/diffraction (SLS)
- too much wave energy enters the harbour due to overtopping (SLS)
- the breakwater collapses (ULS) and too much wave energy enters via the breach Here the first cause is mentioned but this is decided by the harbour lay out and not by the breakwater cross section.
Figure 1: Typical cross section of a vertical breakwater
Throughout this paper only one cross-section of the breakwater has been considered. Thus, all failure mechanisms and probabilities of failure refer to that cross-section only.
2. FAILURE MECHANISMS
The vertical caisson breakwater which will be considered here is placed on a rubble mound foundation (see Figure 1).
The most important failure mechanisms of this structure are: - sliding of the caisson over the rubble foundation; - tilting of the caisson (not realistic; limited by:) - foundation failure consisting of
- landward sliding of the rubble mound; - landward sliding of the subsoil; - seaward sliding of the rubble mound; - seaward sliding of the subsoil; - excessive settlements.
- changes to the geometry of the foundation at sea or land side by - erosion of the toe of the mound;
- erosion of the subsoil at the sea floor;
- loss of material from the rubble foundation (filter failure); - breach of the wall of the caisson (front side);
- breach of the floor of the caisson;
An overview of these failure mechanisms is given in Figure 2. Differential settlements are not included because this requires the comparison of two caissons.
3. FAULT TREE
In the fault tree the loss of the main function is designated as the top event: in-tranquil harbour Hs-harbour> 0.20m.
This may result from one of three failure mechanisms: - wave energy enters by refraction/diffraction (SLS) - wave energy enters due to overtopping (SLS)
- wave energy enters via a breach of the breakwater(ULS)
Wave overtopping occurs if the daily wave climate is higher than expected or if the crest level is lower than expected. The percentage of time that overtopping causes an intranquil harbour can be probabilistically assessed (see e.g. van Gelder, 1997).
However due to excessive settlement or loss of material from the mound (filter failure) the crest level may be lower than expected.
The sliding of a caisson will cause a breach in the breakwater.
Failure of the foundation of the caisson will also result in a breach. The various failure mechanisms of rubble and sub-soil contribute to this failure probability:
- landward sliding of the rubble mound; - landward sliding of the subsoil; - seaward sliding of the rubble mound; - seaward sliding of the subsoil;
The likelihood of the occurrence of these mechanisms increases if the geometry of the foundation has been corrupted by erosion at the sea or the landside.
Structural failure of the caisson can also induce a breach. The parts most heavily loaded by storm waves are the prime candidates to fail structurally.
The structural failure of the walls and the bottom slab may also lead to the loss of ballast sand and consequently to sliding failure.
Here the question arises of the concrete slabs will fail directly or via the indirect and more time consuming route: loading->cracking->seawater ingress->corrosion of reinforcement->structural failure.
Figure 3b: The fault tree of a caisson breakwater (branch overtopping)
4. SYSTEM FAILURE PROBABILITY
The goal of the construction of the fault tree is twofold. Firstly insight is gained in the interaction between the parts of the breakwater and their failure mechanisms.
Secondly the probability of system failure can be calculated or approximated.
For the second part of the goal the failure probabilities of the mentioned mechanisms have to be calculated by means of Level II or Level III calculations (Burcharth, 1995 or Haile 1996). In these calculations the change of geometry by preceding mechanisms like erosion of the foundation can be included. Simply by calculating the geometry and it's uncertainty at the end of the planning period or more refined as a function of time.
Failure of one of the main mechanisms will lead to the failure of the total system, because the system is a series system. The lower and the upper limits of the system failure probability P(F) are given as
max P(Fi) P(F) P(Fi).
In many practical cases these bounds are sufficiently narrow as one of the failure mechanisms has a probability of failure that is an order of magnitude higher than the others (Burcharth, 1995, the Mutsu Ogawara case or Haile,1996).
For an exact calculation of the system failure probability the dependence between the mechanisms has to be taken into account. Most mechanisms contain Hs and some share resistance variables like the weight of the caisson. These joint variables are responsible for the dependence and reduce the system failure probability below the upper limit.
ACKNOWLEDGEMENT
This work was partially funded by Commission of the European Communities under the contract: MAS3-CT95-0041. Discussions with scientists from Rijkswaterstaat, Ministry of Public Works and the Municipality of the city of Rotterdam have contributed to the clarity of the content of the paper.
REFERENCES
BURCHARTH,H.(1995): Application of reliability analysis for optimal design of monolithic vertical wall breakwaters, Proceedings International Conference on Coastal and Port
Engineering, Rio de Janeiro
VAN GELDER,P.ET AL. (1997): Probabilistic analysis of wave transmission due to overtopping of vertical breakwaters, Internal paper MAST-III PROVERBS
HAILE, A (1996): Probabilistic Design of Vertical Breakwaters, Master thesis Delft
John Dalsgaard Sørensen & Hans F. Burcharth Aalborg University
Sohngaardsholmsvej 57, DK-9000 Aalborg Denmark
1. Introduction
This chapter contains a description of the uncertainties and failure modes used in the probabilistic calculations within PROVERBS. The description is for the geotechnical failure modes based on the limit state equations in the Task II report, chapter 6: (Jacobsen et al. 1999) and the reports (Christiani 1998) and (Sørensen et al. 1996). The structural failure modes are based on the Task III report on limit state equations, (Crouch, 1999). For each failure mode a limit state function is formulated such that the reliability can be estimated using probabilistic methods, as described in e.g. (Madsen et al. 1986).
The following limit equations are formulated (the name in capital letters refers to the name of the implemented FORTRAN subroutine):
• Sliding:
1 Sliding between caisson and bedding layer (SLIDTAK), see (1) in figure 1 and section 3.1
• Failure in rubble mound:
2 Sliding between caisson and rubble foundation (SLIDFRIC), see (1) in figure 1 and section 3.1
3 Failure in rubble mound – straight rupture line (RUBSAND2), see (2) in figure 1 and section 3.2
4 Failure in rubble mound – curved rupture line (RUBSAND4), see (4) in figure 1 and section 3.3
11 Failure in rubble mound, constant volume (RUBUND1), see (10) in figure 1 and section 3.9
• Failure in rubble mound and sand subsoil:
5 Failure in rubble mound and sliding along top of subsoil (RUBSAND3), see (2) in figure 1 and section 3.3
6 Failure in rubble mound and sand subsoil (RUBSAND5), see (5) in figure 1 and section 3.5
7 Failure in rubble mound and sand subsoil (RUBSAND6), see (6) in figure 1 and section 3.6
12 Failure in rubble mound and subsoil, constant volume (RUBUND2), see (9) in figure 1 and section 3.9
• Failure in rubble mound and clay subsoil:
8 Failure in rubble mound and sliding along top of subsoil (SLIDCLAY), see (3) in figure 1 and section 3.3
9 Failure in rubble mound and clay subsoil (RUBCLAY8), see (7) in figure 1 and section 3.7
10 Failure in rubble mound and clay subsoil (RUBCLAY9), see (8) in figure 1 and section 3.8
13 Failure in rubble mound and subsoil, constant volume (RUBUND2), see (9) in figure 1 and section 3.9
• Structural failure modes:
14 Flexural failure in a reinforced concrete beam, see section 3.10 15 Shear failure in a reinforced concrete beam, see section 3.11 16 Cracking limit state function, see section 3.12
17 Initiation of corrosion due to chloride ingress in a reinforced concrete structure, see section 3.13
• Other failure modes:
18 Flexural failure in a reinforced concrete beam, see section 3.14 An overview of the geotechnical failure modes is given in the figure 1.
In some of the failure modes one or two parameters describing the geometry of the failure mechanisms have to be obtained solving an optimization problem where the ratio be-tween the stabilizing and driving forces / moments is minimized. When the failure mode is used as a limit state function in a reliability function this is equivalently obtained by minimizing the value of the limit state function with given values of the stochastic vari-ables.
The bearing capacities related to the geotechnical failure modes described in this report are generally estimated using the upper bound theorem of classical plasticity theory where an associated flow rule is assumed. However, the friction angle and the dilation angle for the rubble mound material and the sand subsoil are usually different. Therefore, in order to use the theory based on an associated flow rule, the following reduced effective friction angle ϕd is used, see (Hansen 1979):
tan sin ' cos sin ' sin ϕ ϕ ψ ϕ ψ d = − 1 (1) where
ϕ' is the effective friction angle and ψ is the dilation angle.
The wave load generated horizontal force on the caisson and the vertical uplift are esti-mated using the Goda formulae. The wave load also generates a pore pressure in the rub-ble mound and in the subsoil. The horizontal component on the rupture boundary of this pore pressure has to be taken into account when considering the various failure mecha-nisms. The approximate model shown in (Christiani 1998) can be used to estimate the horizontal force FHU.
2. Reliability analysis
The geotechnical failure modes described in section 2 can be characterized as ultimate limit states and the probability of failure within the design lifetime T can then be esti-mated by Pf S P
{
gi}
i n = − = ⎛ ≤ ⎝⎜ ⎞ ⎠⎟ = Φ( β ) ( )X 0 1 (2)where gi( )X is the safety margin for failure mode no i and X is a vector with the sto-chastic variables. βS is the system reliability index corresponding to the probability of failure Pf estimated by FORM analysis, see (Madsen et al. 1986). Φ( )⋅ is the distribu-tion funcdistribu-tion for a standard normal distributed stochastic variable. The probability of fail-ure Pf
i for each failure mode can also be determined by FORM:
Pf P g
(
i)
P i iT ii = ( )X ≤0 ≈ (β α− U≤0)=Φ(−β ) (3)
where βi is the reliability index for failure mode i , αi is a unit vector with elements indicating the relative importance of the stochastic variables and U is a vector with stan-dardized Normal distributed stochastic variables.
The probability of failure for the system can then be approximated by
Pf ≈Φn
(
β;ρ)
(4)where β =(β1,...,βn) and the correlation coefficient ρ is determined byij
ρij =αiTαj (5)
The n -dimensional normal distribution function Φn
( )
is numerically evaluated using the Hohenbichler approximation or Ditlevsen-bounds are calculated, see (Madsen et al. 1986).The systems reliability index β is defined bys
βs =−Φ−1
( )
Pf (6)Alternatively the probability of failure for the single failure modes and for the system can be estimated using simulation. This is especially relevant when the limit state equations are discontinuous.
For a given kinematically admissible failure mechanism the internal work for an infini-tisimal displacement δ = 1 is denoted WI i, ( ,θ X where θ is a vector with the free pa-) rameters describing the mechanism. Correspondingly the externnal work for the infini-tisimal displacement is denoted WE i, ( , )θ X .
The limit state function is written
gi( )X =min
{
WI i, ( , )X −WE i, ( ,X)}
θ θ θ (7)
where the minimization of WI i, ( ,θ X -W) E i, ( , )θ X is performed with respect to θ . Fur-ther, constraints can be added to (7) in order to limit the displacements and to describe the displacements of the failure mechanism.
The sensitivity of the reliability index and the probability of failure can be measured in several ways:
1) By the α -vectors obtained by a FORM analysis, see (3). α indicates the relativei2 importance on the reliability index β of stochastic variable no. i .
2) By the derivatives dp dβi and dp dβs
which give derivative of the reliability index or the systems reliability index with respect to a parameter p . p could be a deterministic parameter or it could be the expected value or the standard deviation of a stochastic variable. dp dβi and dp dβs
can easily be obtained on the basis of FORM/SORM analy-ses.
3) If the reliability analysis is performed using simulation methods a sensitivity measure which indicates the importance of the stochastic variables is:
∑ ⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = = n j j i i d 1 2 σ β σ β (8)
The limit state equations described in section 3 are implemented in a computational reli-ability program with the following submodules, see definition of names for the limit states in section 1.:
PRADSR: reliability analysis of vertical wall breakwater for individual limit states and for systems using FORM/SORM or simulation methods.
-BETACA: calculation of probability of failure and reliability index for one limit state
-FAILEL: calculation of value of limit state equation for given realiza-tion of stochastic variables.
-FORMOCA: calculation of pulsating wave load using GODA formulae.
-IMPACT: calculation of impact wave load using PROVERBS formulae
-SLIDTAK: limit state function for sliding between caisson and bedding layer
-SLIDFRIC: limit state function for sliding between caisson and rubble foundation
-RUBSAND2: limit state function for failure in rubble mound -RUBSAND3: limit state function for failure in rubble mound and
sliding along top of sand subsoil
-RUBSAND4: limit state function for failure in rubble mound -RUBSAND5: limit state function for failure in rubble mound and
sand subsoil
-RUBSAND6: limit state function for failure in rubble mound and sand subsoil
-RUBCLAY8: limit state function for failure in rubble mound and clay subsoil
-RUBCLAY9: limit state function for failure in rubble mound and clay subsoil
-RUBUND1: limit state function for constant volume failure in rubble mound
-RUBUND2: limit state function for constant volume failure in rubble mound and subsoil
3. Limit state equations
This section contains a short description of the limit state equations used for the reliability analyses in PROVERBS. The geotechnical limit state equations are described in detail in Vol. IIb, section 6, (Jacobsen et al. 1999).
The following general symbols are used in the limit state equations: FG weight of caisson reduced for buoyancy
FU wave induced uplift FH horizontal wave force
FHU horizontal pore pressure force ϕd
ωiV vertical displacement of rupture zone i ωiH horizontal displacement of rupture zone i γr specific weight of rubble material
γ w specific weight of water cu undrained shear strength Ai area of zone i
lCF length between point C and F
rAB length between point A and B (=length of radius) Wi internal work in zone i
Z model uncertainty for the uncertainty related to a given geotechnical failure mode
3.1 Sliding between caisson and bedding layer/rubble foundation (SLIDTAK, SLIDFRIC)
Figure 2. Sliding failure between caisson and bedding layer/rubble foundation.
The failure mechanism is horizontal sliding on the bedding layer. The limit state func-tion is written:
g =(FG −FU) tanμ−FH (9)
where:
tanμ = friction coefficient f if sliding occurs between concrete base plate and the
bed-ding layer = ω1 ϕ
1
Figure 3. Displacement field.
3.2 Failure in rubble mound (RUBSAND2)
Figure 4. Failure in rubble mound.
The effective width Bz of the caisson is determined such that the resultant vertical force
FG −FU is placed Bz / 2 from the heel of the caisson, see figure 4.
The failure mechanism is a unit displacement along the line AB. The limit state equation is obtained by solving an optimization problem where the optimization variable is:
θ angle of rupture line and one constraint is used: 0≤ ≤ 1 + + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − θ tan h B a b II z
: rupture line should be in the rubble mound
The limit state function is written:
3.3 Failure in rubble mound and sliding along top of subsoil (clay / sand) (RUB-SAND3, SLIDCLAY)
Figure 5. Failure in rubble mound.
The failure mechanism is a unit displacement δ = 1 along top of subsoil. The limit state function for sand subsoil is written:
( ) tan ( )tan ( ) 2 2 1 d G U d H HU w r A F F F F Z g = γ −γ ϕ + − ϕ − + (11)
The limit state function for clay subsoil is written:
g =ZlBCcu −(FH +FHU) (12)
3.4 Failure in rubble mound (RUBSAND4)
The failure mechanism is a unit displacement δ = 1 along line AB. The limit state equa-tion is obtained by solving an optimizaequa-tion problem where the optimizaequa-tion variables are: θ angle of rupture line
α angle of zone 2
and the following constraints are added: 0≤θ
0≤ ≤α θ
β α θ+ − > 0 A3 ≥0
The limit state function is written
g =Z(W1+W2 +W3)+(FG −FU)ω1V −(FH +FHU)ω1H (13)
3.5 Failure in rubble mound and sand subsoil (RUBSAND5)
Figure 7. Failure in rubble mound.
The failure mechanism is sliding along line AB. The limit state equation is obtained by solving an optimization problem where the optimization variable is:
θ angle of rupture line
and the following constraints are added: tan− + + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ 1 h B a b II z
θ rupture line should enter the subsoil
θ1 π ϕ
2 1
≥ − d
The limit state function is written:
g =Z(W1+W2 +W3 +W4)+(FG −FU)ω1V −(FH +FHU)ω1H (14)
3.6 Failure in rubble mound and sand subsoil (RUBSAND6)
Figure 8. Failure in rubble mound.
The failure mechanism is sliding along line AB. The limit state equation is obtained by solving an optimization problem where the optimization variable is:
θ angle of rupture line
and the following constraints are added: tan− + + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ 1 h B a b II z
θ rupture line should enter the subsoil
θ1 π ϕ
2 1
≥ − d
A2 ≥0
The limit state function is written:
3.7 Failure in rubble mound and clay subsoil (RUBCLAY8)
Figure 9. Failure in rubble mound and in clay subsoil.
The failure mechanism is a unit displacement δ = 1 along the line BC. The limit state equation is obtained by solving an optimization problem where the optimization variable is:
θ angle of rupture line
The following constraints are added: θ ≥ 0 θ ≥ ϕ + + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − − tan 1 1 h B a b II z d
If cu is modelled as a stochastic variable the limit state function is:
g =Z(W1+W2 +W3 +W4 −W5)−(FG −FU)sinθ −(FH +FHU)cosθ (16) where the internal work from rupture along BC is
W cu s ds
lBC
1 0
= ∫ ( ) (17)
where c su( ) is the undrained shear strength of clay as function of distance s . The internal work from rupture along CD is
W cu s ds rCF 2 0 4 = ∫ + ( ) (π θ) (18)
The internal work from rupture along DE is W cu s ds lDE 3 0 = ∫ ( ) (19)
The internal work from Prandl rupture zone 2 is
W c s dsd r u CF 4 0 4 = ∫ ∫ + 0 ( , ) ( ) τ τ π θ (20)
If cu is modelled as a stochastic field the limit state function is:
g =Z(μW +UσW −W5)−(FG −FU)sinθ −(FH +FHU)cosθ (21) where
U is a standard normal distributed stochastic variable : N(0,1)
μW is the expected value of W =W1+W2 +W3+W4. Determined by numerical inte-gration using the trapez rule.
σW is the standard deviation value of W =W1+W2+W3+W4. Determined by nu-merical integration using Monte Carlo simulation and σW i j
j i Cov W W 2 1 4 1 4 = ∑ ∑ = = [ , ]
where Cov W W[ i, j] is the covariance of Wi and Wj.
As examples the expected value E[W1] and the covariance Cov[W1,W3] are:
= ∫ BC l u s ds c E W E 0 1] [ ( )] [ (22) 2 0 1 2 1 0 3 1, ] [ ( ), ( ) [W W Covc s c s dsds Cov BC DE l u u l ∫ ∫ = (23)
where E[cu(s)] is the expected value of cu(s), see section 3 and Cov[cu(s1),cu(s2)] is the covariance function of cu(s) at the positions corresponding to s and 1 s .2
3.8 Failure in rubble mound and clay subsoil (RUBCLAY9)
Figure 10. Failure in rubble mound and in clay subsoil.
The failure mechanism is a unit rotation β = 1 about point D. The limit state equation is obtained by solving an optimization problem where the optimization variables are:
xD x-coordinate of point D yD y-coordinate of point D The following constraints are added: yD ≥ 0 Bz xD Bz a b 2 ≤ ≤ + + rBDcosα = yD +hII α ≥ 0 θ ≥ 0
If cu is modelled as a stochastic variable the limit state function is:
g Z Z W W Z FG FU xD Bz) (FH FHU)yD 2 1 )( ( ) (− 1− 2 + 3 + 4 − − − − + = (24)
where the internal work from rupture along circle BC is:
W rBD cu s ds rBD 4 0 2 = α∫ ( ) (25)
If cu is modelled as a stochastic field the limit state function is: g Z W W W W U W FG FU xD Bz) (FH FHU)yD 2 1 )( ( ) ( 4 4 3 2 1 − + + + − − − − + − = μ σ (26) where
U is a standard normal distributed stochastic variable : N(0,1) μW
4 is the expected value of W4. Determined by numerical integration using the trapez
rule. σW
4 is the standard deviation value of W4. Determined by numerical integration using
Monte Carlo simulation.
3.9 Failure in rubble mound – constant volume (RUBUND1)
Figure 11. Failure in rubble mound – constant volume.
The failure mechanism is a circular rupture boundary with a unit rotation β = 1 about point D. The limit state equation is obtained by solving an optimization problem where the optimization variables are:
xD x-coordinate of point D yD y-coordinate of point D The following constraints are added: yD ≥ 0
Bz xD Bz a b
2 ≤ ≤ + +
rBDcosα = yD +hII
c≥hII
g Z W W W FG FU xD Bz) (FH FHU)yD 2 1 )( ( ) ( 1+ 2 + 3 − − − − + = (27)
where the internal work from rupture along circle AB is
= ∫ BD r u BD c s ds r W θ 0 1 3 ( ) (28)
and cu1( ) is the undrained shear strength of rubble mound material as function of dis-s
tance s
If cu is modelled as a stochastic field the limit state function is:
g Z W W W U W FG FU xD Bz) (FH FHU)yD 2 1 )( ( ) ( 3 3 2 1+ + + − − − − + = μ σ (29) where
U is a standard normal distributed stochastic variable : N(0,1)
3
W
μ is the expected value of W . Determined by numerical integration using the trapez3
rule.
3
W
σ is the standard deviation value of W . Determined by numerical integration using3
Monte Carlo simulation.
3.10 Failure in rubble mound and subsoil – constant volume (RUBUND2)
The failure mechanism is a circular rupture boundary with a unit rotation β = 1 about point D. The limit state equation is obtained by solving an optimization problem where the optimization variables are:
xD x-coordinate of point D yD y-coordinate of point D The following constraints are added: yD ≥ 0 Bz xD Bz a b 2 ≤ ≤ + + rBDcosα = yD +hII α ≥ 0 θ ≥ 0
If cu is modelled as a stochastic variable the limit state function is:
g Z W W W W FG FU xD Bz) (FH FHU)yD 2 1 )( ( ) (− 1 − 2 + 3 + 4 − − − − + = (30)
where the internal work from rupture along circle AC is
W rBD cu s ds c s ds r u r BD BD 4 1 0 0 2 2 = ⎛ ∫ + ∫ ⎝ ⎜ ⎞ ⎠ ⎟ ( ) ( ) θ α (31) where cu s
1( ) undrained shear strength of rubble mound material as function of distance
s
cu2( )s undrained shear strength of subsoil as function of distance s If cu is modelled as a stochastic field the limit state function is:
g Z W W W W U W FG FU xD Bz) (FH FHU)yD 2 1 )( ( ) ( 4 4 3 2 1 − + + + − − − − + − = μ σ (32) where
U is a standard normal distributed stochastic variable : N(0,1) μW
4 is the expected value of W4. Determined by numerical integration using the trapez
rule. σW
4 is the standard deviation value of W4. Determined by numerical integration using
3.11 Flexural failure of reinforced concrete beam The limit state function is written:
g d f f f pL s y s y c = ⎛ − ⎝ ⎜ ⎞ ⎠ ⎟ − ρ 2 ρα 2 1 0 4 0 8 0 086 . . . (33) where
ρs is the ratio of the steel reinforcement area over the area of the concrete cross-section
d is the depth of section from compression face to centre of tensile steel reinforce-ment
fy yield strength of steel reinforcement fc uniaxial compressive strength of concrete
p uniformly distributed pressure action from wave load L beam span between walls
α coefficient which takes into account long-term effects on the compressive strength, usually α =0.85.
3.12 Shear failure of reinforced concrete beam The limit state function is written:
g =d
(
(
1 6. −d)
(
1 2. +40ρl)
0 0525.( )
fc 2 3/ +015. σcp)
−0 6. pL (34) wherefc is the uniaxial compressive strength of concrete
d is the depth of section from compression face to centre of tensile steel reinforce-ment
ρl is the lesser of the longitudinal tension reinforcement ratio and 0.02
σcp = N A/ g where N is the design axial force and Ag is the gross area of the cross section
p uniformly distributed pressure action from wave load L beam span between walls
3.13 Cracking limit state function
g f f l sr s = − ⋅ ⎛ + ⎝ ⎜ ⎞ ⎠ ⎟ −⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ − 0 3 2 91 10 50 10 1 3 2 . . φ ρ (35) where
φ is the bar diameter (in mm)
ρt is the effective reinforcement ratio
fsr is stress in tension steel at which the load causes crack initiation fs is the stress in tension steel under the design loading
3.14 Initiation of corrosion due to chloride ingress in a reinforced concrete structure For analysis of existing concrete structures the following limit state function can be used to model initiation of repair / maintenance at time t in a given point j of the structure
gj = −t TI −TR (36)
where TI is the time until initiation of corrosion and TR is the time from initiation of corrosion until repair actions have to be made. The limit state equation is described in detail in annex A.
3.15 Scour failure for circular roundheads on sand
The limit state function is written, see (Sørensen & Burcharth 1998) and (Sumer et al. 1996): (no rubble foundation and bedding layer)
= −0.5A
(
1−exp(
−0.175(
KC−1)
)
)
B S
g (37)
where S is the scour depth, B is the caisson width, A models the model uncertainty (N(1, 0.6)) and the Keulegan Carpenter number KC is determined from
B T U KC = m p (38)
(
s p)
p S m L h T H U / 2 sinh 1 π π = (39)g s H T p S p π 2 = (40) S
H is the significant wave height, s is the wave steepness and p h is the water depth.s
The wave length L is determined fromp
(
s p)
p p h L T g L tanh2 / 2 2 π π = (41)3.16 Hydraulic instability of foundation rubble mound armour layer
The limit state function is written (Madrigal et al. 1995) and (Sørensen & Burcharth, 1998): S od s n N H h h D A g − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − Δ = 0.19 60 . 0 ' 8 . 5 (42)
where A is the model uncertainty (N(1, 0.6), Δ=
(
ρarmour /ρwater)
−1, h' is the water depth to caisson bottom, h is the water depth in front of toe and s Nod is the damage paramter. D is the equivalent cube length of the armour.n4. Stochastic models
Figure 13. Wave induced quasi-static load according to (Goda 1972 and 1974). Wave load
The main load for vertical wall structures is due to wave loading. Depending on the ge-ometry of the rubble mound and the caisson the wave loading can be characterized as, see for Task I report:
• Quasi-static (pulsating) wave loads are estimated using the Goda formula, see Task I report. The model uncertainties related to horizontal and uplift wave forces are as-sumed to be fully correlated, see table 1. Also model uncertainties related to horizon-tal and uplift wave moments are assumed to be fully correlated, see table 1.
• Impact loading characterized by a very high load but with a very short duration con-sists of a horizontal and a vertical (uplift) part. They can be estimated by the model described in the Task I report. The model uncertainties related to the impact loads are modeled by the stochastic variables no. 18, 19, 20 and 21, see table 1.
Wave height
The maximum significant wave height HST in the design lifetime T usually has to be modeled on the basis of a limited number N of wave height observations. Here an ex-treme Weibull distribution is used, see Burcharth (1993) and (Sørensen and Burcharth 1998): F h h H u H T S T ( ) exp ' = − ⎛− −⎛⎝⎜ ⎞⎠⎟ ⎝ ⎜⎜ ⎞⎠⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 α λ (43) where λ is the number of observations per year. α , u and H' are parameters to be fitted to the observed data. In order to model the statistical uncertainty u is modeled as a Nor-mal distributed stochastic variable with coefficient of variation
V N u = + + − 1 1 2 1 1 1 2 Γ Γ ( / ) ( / ) α
α . The model uncertainty related to the quality of the meas-ured wave data is modeled by a multiplicative stochastic variable FH
S which is assumed
to be normal distributed with expected value 1 and standard deviation equal to 0.05 or 0.2 corresponding to good or poor wave data, see Burcharth (1993). Further, the water level set-up due to storm wind and waves (storm surge) is difficult to estimate except for sim-ple conditions (straight coastline and constant slope of sea bed). The uncertainty related to storm surge varies considerably with the environmental conditions. In the reliability analysis this uncertainty on the storm surge water level is modeled by a stochastic vari-able with mean zero and standard deviation equal to 0 05. HS.
If the design significant wave height (e.g. as the 98 % fractile in the one-year maximum wave distribution), the parameters (λ , α and u ) and the number of wave data observa-tions N are known, the parameters in the statistical model for the significant wave height can be determined.
The tidal elevation ς is modeled as a stochastic variable with distribution function, see Burcharth (1993) and (Sørensen and Burcharth 1998):
Fς ς π ς ς ( )= arccos⎛− ⎝ ⎜ ⎞ ⎠ ⎟ 1 0 (44) where ς0 is the maximum tidal height.
Geotechnical parameters
In general the material characteristics of the soil have to be modeled as a stochastic field. The parameters describing the stochastic field have to be determined on the basis of the measurements which are usually performed to characterize the soil characteristics. Since these measurements are only performed in a few points also statistical uncertainty due to the few data points is introduced and have to be included in the statistical model. Further, the uncertainty in the determination of the soil properties and the measurement uncer-tainty have to be included in the statistical model. In the literature the undrained shear strength of clay is often modeled by a log-Gaussian distributed stochastic field
{
cu( , ) , see e.g. (Keaveny et al. 1989) and (Andersen et al. 1992). The expected valuex z}
function E c
[
u( , ) and the covariance function x z]
C c[
u(x1,z1),cu(x2,z2)]
can typically be written, see Burcharth (1993) and (Sørensen and Burcharth 1998):E c
[
u( , )x z] [
=E cu( )z]
(45)where (x z1, 1) and (x2,z2) are two points in the soil. E c
[
u( ) models the expectedz]
value of the undrained shear strength in depth z . C c
[
u(x1− x2,z1 −z2)]
models the covariance between cuat position (x z1, 1) and cu at position (x2,z2). It is seen that the expected value depends on the depth and the covariance depends on the vertical and hori-zontal distances. Generally the correlation lengths in horihori-zontal and vertical direction will be different due to the soil stratification. The statistical parameters describing E c[
u( )z]
and C c
[
u(x1−x2,z1−z2)]
should be modeled using Bayesian statistics such that prior, subjective knowledge on the values of the parameters can be combined with measure-ments from the actual site, see e.g. (Lindley 1976). In practical calculations the stochastic field are discretized taking into account the correlation lengths of the field. If an integral over some domain is used, the expected value and the standard deviation of this integral can be evaluated numerically.Since the breakwater foundation is made of friction material and it is assumed that foun-dation failure modes can develop both in the rubble mound and in sand subsoil, statistical models for the effective friction angle and the angle of dilation are needed for the rubble material and the sand subsoil. In this paper these angles are modeled by Lognormal sto-chastic variables, i.e. the spatial variation is not taken into account.
Model uncertainty connected to the mathematical models for the geotechnical failure modes used to estimate the soil strength can be important due to the relatively high un-certainty related to the models used. If slip failure models based on the upper bound theo-rem of plasticity theory are used these can be evaluated by comparison with results from more refined numerical calculations using nonlinear finite element programs with realistic constitutive equations for the soil implemented. Estimates of the model uncertainties can then be obtained by comparing the comparing the results. The estimates of the model un-certainties should also to some degree depend on professional, subjective insight in the failure modes considered.
For clay subsoil the mean value function and covariance function are assumed to be mod-eled by
E c
[
u( , )x z]
=cu0+c zu1 (47)C c
[
u x z cu x z]
c(
cz z)
(
c(
x x)
)
u
( 1, 1), ( 2, 2) =σ2 exp −α 1− 2 exp⎝⎜⎛− β 1− 2 2⎞⎠⎟ (48) where cu0 and cu1model the expected value, σc
u is the standard deviation and αc and
βc model the correlation.
In cases where no detailed information on the subsoil is available the following parame-ters for strong and weak subsoil can be used:
Undrained subsoil (clay): Weak clay: c = 79 kPa and u0 c = 1 kPa/m. Strong clay: u1 c =u0
173 kPa and c = 0 kPa/m. Further, the following parameters can be used : u1 u
c
σ =30 kPa, α =0.33 1
m− and β =0.033 m−1.
Drained subsoil (sand): Weak sand: E[ϕ2]=39.0°, E[ψ2]=10.2°. Strong sand: ] [ϕ2 E =42.8°, E[ψ2]=15.3°. Coefficient of variation = 10 %. i Xi Mean Standard Deviation Distribu-tion 1 HS significant wave height [ m ] see above see above W
2 u Weibull parameter [ m ] 2 see above N
3 FH
S model uncertainty on wave height 1 0.05/0.20 N
4 sM wave steepness factor 1 0.25 N
5 ς tidal elevation, maximum ς0 = 0.8 m Cosine
6 UF
H model uncertainty horizontal force 0.90 0.20 / 0.05 N
7 UF
U model uncertainty uplift 0.77 0.20 / 0.05 N
8 UM
H model uncertainty horizontal moment 0.81 0.40 / 0.10 N
9 UM
U model uncertainty uplift moment 0.72 0.37 / 0.10 N
10 ρc average density of caisson [ t m/ 3] 2.23 0.11 N
11 ϕ1 effective friction angle - rubble mound 46° 4.6° LN 12 ψ1 angle of dilation – rubble mound 16.7° 1.67° LN 13 ϕ2 effective friction angle - sand subsoil 39.0°/42.8°
weak/strong
3.9° / 4.3° weak/strong
LN 14 ψ2 angle of dilation – sand subsoil 10.2°/15.3° 1.0° / 1.5° LN
15 U clay strength 1 0 N
16 f Friction coefficient 0.636 0.0954 LN
17 cu undrained shear strength for impact load 400 kPa 80 kPa LN
18 k factor for impact load 0.086 0.084 LN
19 c factor for impact load 2.17 1.08 LN
20 R model uncertainty factor for impact rise time
1 0.3 LN
21 UI model uncertainty factor for impact forces
1 0.5 LN
22 fy yield stress for steel [kPa] 360 18 LN
23 fc compressive strength for concrete [kPa] 30 4.5 LN Table 1. Statistical model. W: Weibull, N: Normal, LN: LogNormal.
Structural quantities:
The yield stress for steel and the compressive strength for concrete are often modeled by Lognormal distributed stochastic variables.
The complete list of stochastic variables used in the probabilistic calculations and exam-ples of parameter values are indicated in table 1.
5. References
Andersen, E.Y. and B.S. Andreasen and P. Ostenfeld-Rosenthat (1992). Foundation Re-liability of Anchor Block for Suspension Bridge. Proc. IFIP WG7.5, Lecture notes in Eng. Vol. 76, Springer Verlag, pp. 131-140.
Burcharth, H.F. (1993). Development of a partial safety factor system for the design of rubble mound breakwaters. PIANC PTCII Working group 12, Subgroup F, final report. Published by PIANC, Brussels.
Christiani, E. (1998): Application of reliability in breakwater design. Ph.D. thesis, De-partment of Civil Engineering, Aalborg university.
Crouch, R. (1999): Structural aspects. PROVERBS – final report, Volume IIc.
Goda, Y. and Fukumori, T. (1972). Laboratory investigation of wave pressures exerted upon vertical and composite walls. Coastal Engineering in Japan, Vol. 15. pp. 81-90. Goda, Y. (1974). A new method of wave pressure calculation for the design of composite breakwater. Proc. 14th Int. Conf. Coastal Eng., Copenhagen, Denmark.
Hansen, B. (1979): Definition and use of friction angles. Proc. Int. Conf. VII ECSMFE, Brighton, UK.
Jacobsen, K.P. & J.D. Sørensen & H.F. Burcharth & L.B. Ibsen (1999): Failure modes – Limit state equations for stability. PROVERBS – final report, Volume IIb, chapter 6, an-nex A.
Keaveny, J.M. and F. Nadim and S. Lacasse (1989). Autocorrelation Function for Offsho-re Geotechnical Data. Proc. ICOSSAR’89, pp. 263-270.
Lindley, D.V. (1976). Introduction to Probability and Statistics from a Bayesian View-point, Vol 1+2. Cambridge University Press, Cambridge.
Madrigal, B.G. and J.M. Valds (1995). Results on stability tests for rubble foundation of a composite vertical breakwater. MAST II/MCS, CEPYC-CEDEX, Madrid.
Madsen, H.O., S. Krenk & N.C. Lind (1986): Methods of Structural Safety. Prentice-Hall. Sumer, B.M. and J. Fredsøe (1996). Scour at the head of a vertical-wall breakwater. Coastal Engineering.
Sørensen, C.S. & L.B. Ibsen & F.R. Jakobsen & A. Hansen & K.P. Jakobsen (1996): Bearing capacity of caisson breakwaters on rubble mounds. Annex IX of Foundation de-sign of caisson breakwaters. Publication no. 198. Norwegian Geotechnical Institute, Oslo. Sørensen, J.D. and H.F. Burcharth. (1998). Implementation of safety in the design. Report of Sub-group D, PIANC Working Group 28.
Annex A. Reliability Analysis of Existing Concrete Caissons subject to
Chloride Ingress
In this annex a method is presented for estimating the reliability of existing concrete structures subject to chloride ingress. The method can be used for vertical wall breakwa-ters made of reinforced concrete. Chloride ingress is one of the most common destructive mechanisms for this type of structures. The most typical type of chloride initiated corro-sion is pitting corrocorro-sion which may locally cause a substantial reduction of the cross-sectional area and cause maintenance and repair actions which can be very costly. Further, the corrosion may make the reinforcement brittle, implying that failure of the structure might occur without warning.
Probabilistic analysis of the time to initiation of corrosion in concrete structures has been treated in a number of papers, see e.g. (Engelund 1997), (Pedersen & Thoft-Christensen 1993), (Poulsen 1995) and (Hoffman & Weyers 1996). Using different models of chloride ingress and different probabilistic models these researchers all determine the probability of corrosion at an arbitrary point of a structure. The probability of corrosion can be used to take decisions on possible maintenance and repair actions, see e.g. (Engelund & Søren-sen 1998).
The owners of the structures, therefore, have to plan and perform measurements which can be used as a basis for an assessment of the structure and as a basis for deciding which repair and maintenance strategy should be applied. The experiments and maintenance strategies can be very costly and careful planning is therefore needed. The optimal deci-sions about the experimental plan and the maintenance methods can be made on the basis of modern structural reliability theory together with economic decision theory, see e.g. (Kroon 1994) and (Raiffa & Schlaifer 1961).
A1. Stochastic modeling
Chloride is transferred from the surroundings into the concrete through the system of capillary pores and microcracks. The transport is a complex phenomenon involving vari-ous mechanisms such as diffusion of chloride and transport of chlorides by the flow of water. However, measurements from existing uncracked structures (cracks smaller than 0.1 mm, see for example (Tuutti 1982)) with not too low w/c-ratios support the assump-tion that the chloride concentraassump-tion can be considered as the soluassump-tion to a suitable one-dimensional linear diffusion problem (Fick's second law of diffusion) which can be writ-ten ∂ ∂ ∂ ∂ ∂ ∂ c x t t x D x t c x t x ( , ) ( , ) ( , ) = ⎛⎝⎜ ⎞⎠⎟ , c( , )0t =cS (A1)
where c x t( , ) is the chloride concentration at depth x at the time t. D x t( , ) is the effec-tive transport (diffusion) coefficient and cS is the surface chloride concentration. Eq. (A1) can be solved numerically, see e.g. (Engelund et al. 1995) and (Engelund 1997). It is important to notice that it is not assumed that chloride penetration is only caused by diffusion. The diffusion equation has been selected because the solution can be brought to fit measurements from existing structures and because the parameters in the model can be given a physical interpretation. Any other model which can be brought to fit the measured chloride concentrations can in principle be implemented. However, it can be difficult to gain any physical understanding of the problem if the parameters in the model have no physical meaning.
The diffusion coefficient and the surface concentration are both modeled as stochastic variables. However, these variables also exhibit a random spatial variation. The random spatial variation of the diffusion coefficient is caused by the spatial variation of the com-pression of the concrete and the variation of the w/c-ratio. The random variation of the surface concentration is a natural effect of the variation of the spray of salt-containing water. The surface concentration may also exhibit a systematic variation. E.g. for piers in a marine environment the surface concentration depends on whether the surface faces up-stream or down-up-stream and on the distance to the mean water level.
The cover thickness d which is defined as the distance from a point on the surface of a structure to the reinforcement, is modeled as a stochastic variable. The cover thickness will due to the variation in the placing of the reinforcement also exhibit a spatial varia-tion. Corrosion is assumed to start when the chloride content at the depth of the rein-forcement exceeds a critical value ccr which also is modeled as a stochastic variable. The critical value depends on the humidity of the concrete. Because the humidity of the con-crete depends on the permeability of the concon-crete which exhibits a random spatial varia-tion also the critical value exhibits a random spatial variavaria-tion.
Only crude models for propagation of corrosion exist. On this basis it is difficult to for-mulate a probabilistic model. In this paper, therefore, no attempt will be made to formu-late a model which can be used to determine the probability that a given reduction of the reinforcement cross-section has taken place at a given time.
A2. Probabilistic Analysis of Chloride Profiles
The estimation of the time to initiation of corrosion in an existing reinforced concrete structure is usually based on chloride profiles obtained from the structure. A chloride pro-file consists of a number of measurements of the chloride concentration as a function of the distance to the surface, x. Using such chloride profiles the surface concentration and the diffusion coefficient can be estimated. In the following it is assumed that the diffusion
