Computer aided evaluation of the reliability of a breakwater design

FINAL REPORT

CIAO PROJECT GROUP

COMPUTER AIDED EVALUATION

OF THE RELIABILITY OF A

- 2

-C l P ·- G E G EVENS t>; 0 1'-11 N f'a. I J i<r:': 8 11?. L .l 0 THEE h ~ DE N HA A G

Computer.

Compute~ aided evaluation of the ~eliability of a

bceakwate~ design : final ~epo~t I ClAD p~oject g~oup. -Zoete~roeet~: ClAD.-- Ill.

Met l i t . opg.

ISBN 90-6818-019-3

8180 365.2 UDC 681.3.02

T~efw.: compute~ aided design.

G)1985 CIAO Association, Zoetermeer, the Netherlands

All rights reserved

No part of this publication may be reproduced, stored in a computerized filing system and/or published in any form or in any way, be it graphical, electronic, mechanical, by photocopying or in any other manner, without the prior written consent of the publisher.

CIAO accept no responsibility with regard to the content of this report, nor with regard to activities undertaken on the basis of this report.

"" 3

-ClAD: Association for computer applications in applied engineering

Ciad is an independent non~profit~making organisation devoted to the effec-tive use of computer applications in applied engineering. It has been active since 1968. The members of ClAD are consulting engineers, contracting corn~

panies, government departments, educational and research establishments, com-puter centres and software houses. In the Association the members find common interests and aims. By mutual discussion and collaboration they tackle problems that they would otherwise have to face alone.

The activities undertaken by ClAD for its members include:

* collecting and passing on in organised form information about computer software, existing and new working methods, hardware and other matters that relate to efficient use of computers

* establishing and maintaining contacts with, and participating in the ac~ tivities of Dutch and foreign organisations and government departments with similar interests; in this way ClAD becomes the channel of communication and the mouthpiece of all its members,

* stimulating, coordinating and supporting project groups formed by the mem-bers, in such a way as to create for the members a framework for fruitfull colaboration in the field of studies, software development and the promo~ tion of any other common interest within the scope of the aims of the Association.

*carrying on discussions with institutes of regular and private education in order to contribute to the satisfactory training in problems of information processing that are important in applied engineering.

ClAD

P.O,Box 74

2700 AB Zoetermeer, the Netherlands tel +31 (079) 219324

4

-CONTENTS

PREFACE

LIST OF PARTICIPANTS OF THE CIAO PROJECT GROUP BREAKWATERS

INTRODUCTION

1.1 Terms of Reference 1.2 Scope of Work 1.

3

Methodology

2 CONCLUSIONS AND RECOMMENDATIONS

2.1 Conclusions 2.2 Recommendations

3 RELIABILITY THEORY

3.1 Risk analysis for a breakwater 3.2 Probabilistic Calculations

3.3

Application to a breakwater 4 SIMULATION TECHNIQUES 4.1 Computational Models 4.2 Geotechnical Models 4.3 Hydraulic Models

4.4 Modelling Concrete Fracture in Armour Units 4.5 Physical models 5 CASE STUDY page 5 6 7 7 8 9 11 11 1 3 15 1 5 23 26 30 30 30

34

37 38 40

5.1 Objective of a Case Study 40

5.2 The Probability of Failure of a Rubble Mound Breakwater 40

5. 3 Epilogue 46 6 REFERENCES 7 ANNEXES Annex I Annex II Annex III Annex IV

Reliability theory and Fault Trees Geotechnical Aspects

Hydraulic Aspects

Concrete Structural Aspects

47

48 48 62 82 97

~· 5

-PREFACE

With pleasure we herewith submit the report on "Computer Aided Evaluation of the Reliability of a Breakwater Design" achieved as a result of the activities of ClAD. This report has been prepared by a project group including representatives from the Dutch Ministry of Public Works, the Delft Technical University, and a number of civil engineering contracting and consulting firms and research institutes.

The ClAD project group 11 _{Breakwaters" worked in line with the}

recommendations of the PIANC working party on the stability of rubble mound breakwaters which concludes that the subject should be taken forward by individual specialists in certain areas, such as review of existing, selected structures, wave-structure interaction, risk-analysis and construction, maintenance and monitoring.

By focussing on probabilistic design and numerical simulation techniques attention has been given to the fields where the computer may assist in the design process and to new ways for research that are most promising.

We gratefully acknowledge the support of F. Vasco Costa, a member of the Technical Committee on Reliability of Offshore Structures of the ASCE, and a distinguished expert on the subject, who was asked by the group to comment on this manuscript.

We also acknowledge the support of F.C. de Weger International BV who provided data for the case study.

In conclusion, we hope that this study may stimulate the use of probabilistic design methods and that interest in computer aided design of breakwaters may spread far outside the group of participants.

F. B. J. Bar ends Chairman

J. J. van Dijk Reporter

6

-participants of the ClAD project group BREAKWATERS

F.B.J. Barends (chairman,editor) (Delft Soil Mechanics Laboratory) E.O.F. Calle

(Delft Soil Mechanics Laboratory) J.J. van Dijk (reporter)

(Delft University of Technology)

~ J.J. Gallmann

(Delft University of Technology) C.J. van Hassel

(Witteveen en Bos, Deventer) W

.c.

Harden

(Hollandsche Seton~ en Waterbouw, Gouda) F.R. Kalff

(Haskoning BV, Nijmegen)

~ J.K. Kostense

(Delft Hydraulics Laboratory)

L. P.M. Linssen

(DHV, Consulting Engineers, Amersfoort) J.W. van der Meer

(Delft Hydraulics Laboratory) Th. Monnier

(TNO~IBBC, Rijswijk) .., Th. Mulder

(Ballast Nedam Group NV, Amstelveen)

4 J.H. van Oorschot

(AVECO, Rotterdam) J. Vrijling

(Ministery of Public Works, Den Haag)

B.G. Luttikhuizen (project coordinator) (CIAO, Zoetermeer) I Reliability II Geotechnics

III

Hydraulics

IV

Concrete Working Party I II

III IV

X X X X X X X X X X X X X X

. 7

-1.

INTRODUCTION

1.1.

Terms of reference

Against the background of a number of rubble-mound breakwater failures in the last

construction should be various fields.

decenium, the design philosophy in breakwater extended with newly acquired knowledge in

CIAO a Dutch national association stimulating the use of corn~ puters in design processes ~ provided a good opportunity to form a project group "Breakwaters". CIAO formulated the objective of the projectgroup as:

"How can a design of a breakwater be evaluated and optimized through the use of the computer models".

The project group first considered the possibility to compose a general computer program to asses an optimal design. A sequence of design steps should then be formulated using environmental conditions, design formulas, design graphs, design restrictions, material con-stants, execution methods and cost aspects (including repairs and maintenance).

However, in the time available for the project group significant simplifications in such a program would be required, because many design steps can not be programmed with the same sophistication as al~ ready exists in simulation models for some other steps.

Instead, it was considered that the best opportunity to introduce the computer application is in a risk-analysis in the design phase. Risk-analysis improves insight in the reliability of structures by analysing causes of failure and calculating their probability. Because of the complex and iterative character of some of these calculations, the computer is of great use.

Moreover, risk~analysis requires from the designer to present a logical scheme of causes which lead to failure of the structure. Each cause constitutes a failure mechanism for which a simulation model may exist. Blank spots, e.g. failure mechanisms for which no simulation models are available, become apparent.

In order to estimate the probability of failure for a mechanism which lacks a simulation model one can rely on other sources. The

8

-description and definition of relevant failure modes and mechanisms is an important issue for further research.

1.2.

Scope

of work

The risk~analysis offers a method to determine the probability that an

(failure

important function of a or malfunctioning). This

structure is no longer fulfilled probability should be selected having in mind the minimization of the generalized cost of the struc~

ture and the maximization of its utility (Minimax).

Comparison of the contribution of failure mechanisms to the total probability of failure of a design leads the way to rational improve~

ment of a design and to further research directed towards the improve~

ment of weak elements in the design.

In a complex structure like a rubble mound breakwater with a crest wall, many events (accidents) can be distinguished which may lead directly or indirectly to failure of the structure. Hydraulic and geotechnical stability as well as material failure by fracture or loss of position, must be considered. Failure may start in the toe, near the berm, on the slope, at the crest or on the inside slope, and it may progress into total failure with time (accident sequence).

Failure of the structure should be carefully defined. The choice of the definition will predispose the meaning of the calculated prob~

ability of failure. The more precise the definition the clearer the physical meaning and the better the insight presented through the failure probabilities. A clear and detailed definition can be achieved in a fault tree, which in a logical sequence states the possible causes leading to eventual failure.

The CIAO project group limited the scope of work in this report to the following activities:

*

reviewing the reliability theory and risk analysis

*

indicating the simulation models available for the various failure mechanisms

performing a case~study for a rubble mound breakwater, applying the risk~analysis to relevant failure mechanisms and calculating the probabilities of failure using a fault tree approach

9

-*

evaluating results of the case~study.

The underlying theory of the risk~analysis is only presented in outline. Much work is presently done to further develop applications

[7,9,10], In essence it depends on the reader to which degree of sophistication he wants to formulate failure. The project group elaborated an existing design following the probability theory. It would ofcourse have been better to evaluate design alternatives, but this was not possible within the time available.

The case study is the substantial part of the work of the project group. Again, it is only an illustration of how the risk~analysis can be applied, leaving the reader free to simplify or extend the method.

The geometry of the rubble mound breakwater used in the case study and its composition are shown in Figure 1.2.1. Actually, it is an existing

provided by reliability

breakwater, the available information of which has been F.C. de Weger BV, missing data necessary for the study have been fancied as reasonably as possible. Therefore, the outcome of the study cannot be conceived in absolute sense reflecting the reliability of the existing breakwater.

1.3. Methodology

The project group under the chairmanship of Dr.Ir. F.B.J. Barends met several times to formulate the objective, arrange the working party activities and coordinate the progress.

Four working parties were nominated in which various disciplines in~ volved in the study have been covered, to wit:

*

Reliability theory and fault trees

*

Geotechnical stability

*

Hydraulic stability

*

Failure of concrete armour units and crest structure

Fault trees of relevant failure modes have been elaborated for the chosen rubble~mound breakwater by working party I. The other work~ ing parties evaluated the relative probability of failure on the basis

Fault trees of relevant failure modes have been elaborated for the chosen rubble~mound breakwater by working party I. The other work~

ing parties evaluated the relative probability of failure on the basis of selected accident sequences (events) using available simulation models.

If no simulation model was available for a failure mechanism, simple formulas were used describing the physical process and, when no models or formulas were available, probabilities were adopted on the basis of engineering judgement. The contribution of each working party is presented in an annex to this report.

9.0m

0

Quarry run

- 11 ..,

2.

CONCLUSIONS AND RECOMMENDATIONS

2.1. Conclusions

As a result of the study of the project group "Breakwaters" the following conclusions are drawn:

*

The computer is an effective tool for the assessment of the ac~ tual reliability of a breakwater design.

*

Only a few numerical models are at present available to describe failure mechanisms in breakwaters, however, empirically determined design formulas are also useful for probabilistic calculations.

*

Several important processes are not covered by a model or a design formula. White spots are physical models for geotechnical, and mathematical models for hydraulic and structural behaviour.

*

In breakwater design the intuition and experience of the designer and contractor cannot be missed. Each design is related to its loca~ tion with its own specific circumstances with regard to wave loads, earthquake loads, subsoil conditions and building materials. In this respect no attempt seems useful to develop a standard algorithm for which a computer program can be written to contribute in an optimal design proces.

*

The experience of design offices, laboratories and contractors on actual probability distributions of parameters involved is essential for a realistic probabilistic analysis. In general this experience is rather poor and incomplete.

*

'A fault tree represents a useful tool to detect the critical failure mechanisms for geotechnical or hydraulic stability and material strength. The computer allows sensitivity analyses to be made and provides insight in improvements which may be gained by better control of those parameters which can be adjusted during the design or construction stage.

*

The sense. failure

results of a typical case may not be adopted in a general Each situation has a particular fault tree in which a specific mechanism may dominate as a result of a particular initiating event. Probabilities may not be extrapolated to other cases.

*

The evaluation of a study on two alternative designs will show more clearly the power of a probabilistic analysis.

*

For a proper probabilistic analysis more site investigations, laboratory tests, and execution methods are to be copsidered to deter-. mine the probability distribution of the hydraulic, geotechnical and structural parameters, which is the price for an optimal design.

*

The uncertainty in the wave action contributes strongly to the absolute probability of failure of a breakwater. The same holds for the earthquake induced peak acceleration. Although the variance has an endogenic component which cannot be influenced, a computer aided risk analysis may include this aspect and yields the assessment of the actual failure probability feasible.

The case study for a rubble mound breakwater showed that the parameters for which the stability of the breakwater is most sensitive are in subsequent order:

*

Uncertainty in the value of the significant wave height Stability of the armour layer on the slope or berm.

Placement, movement and breakage of armour.

*

Stability of the toe foundation

The approach in the case study is not complete; it is ment as an explanatory example of an application of the probability theory. This holds in particular for the movement and breakage of armour. In struc~

13

-element may cause a chain reaction, not only the variance but also the distribution of the low extreme values of the resistances, the high extreme values of the wave actions (and freak waves), and the obli~

quity and order of arriving waves are required, particularly for large breakwaters.

The probability of failure concerned applies to the final state assuming a rapid propagation of damage once a section fails. In fact, the failure during building stages related to execution methods and the effects of duration of excessive loading to the velocity of spreading of damage is also important. In principle it is possible to assess such problems using model simulations including time dependent effects (dynamic event trees). This aspect requires a great effort and it falls beyond the scope of work of the CIAO project group.

2.2. Recommendations

I It is strongly recommended to improve the present~day state of

the art of breakwater design by adopting a different philosophy and a better methodology based on a probabilistic approach. It is proved by the results of the CIAO project group that a risk~analysis provides a possibility to evaluate safety and economy, and to indicate savings in the long run.

I Control over certain parameters may substantially improve the

reliability by suitable maintenance.

of a design. Reliable parameter values can be guaranteed construction methods, by regular inspection and This also applies to the uncertainty in the significant wave action which can be partly avoided by a proper climatological in~ vestigation, and for phenomena such as rocking and breakage armour. However, extreme wave actions with quite low probability of occurrence are for certain armour the determining factor.

I It is recommended to develop a phenomenological formulation of

... 14 ''

impact and wave overtopping taking into account the local hydro-dynamic forces and air intrusion. Only then a fundamental improvement of design methods for hydraulic stability and structural strength can be established.

*

It is recommended to define relevant wave and earthquake actions for each of the relevant failure mechanisms. The significant wave height for example does not represent the charaqteristic design load~

ing for all phenomena, especially for armour of slender blocks.

*

It is recommended to develop numerical models for those

phenomena, that still lack an adequate physical modelling or design formula, which is particularly the case for several hydraulic and structural aspects.

*

It is recommended to evaluate several existing breakwaters ac~

cording to the presented analysis to provide a sound comprehension of the sensitivity and the importance of component behaviour, and to determine relevant safety factors for various processes involved.

*

It is recommended to provide an opportunity to educate designers and clients in probabilistic methods. The present report, particularly the case study, offers a base for such a course.

*

It is recommended with the knowledge acquired from this study to make a start for the development of an expert system for probabilistic

3.

RELIABILITY THEORY

3.1. Risk analysis for a breakwater

The concepts and terms common in the theory of probability analysis may not be familiar to many engineers. Therefore, a introduc-tion about the probability theory and application is presented in Annex I.

A risk analysis of a complex structure as a breakwater consists of the following steps:

*

Provide a system description that contains sufficient information about the structural system.

*

Analyse the behaviour of the structure if the various structural components fail.

Identify initiating events which may cause failure of components.

*

collect the various accident sequences from initiating event via component failure to total failure of the structure using a fault tree.

*

determine the TOP-event qualitatively by identifying the minimal cut sets.

*

Determine a quantitative estimate of the reliability of the structure by combining the probabilities of component failures and in-itiating events in a minimal cut set according to the logical interaction presented in the fault tree.

The fault tree is a scheme that visualizes a system failure i.e. the occurrence of a selected TOP~event which is identified as total failure or malfunctioning of the structure, or as exceedence of ex-pected generalized costs. It is related to subsystem failures, component failures and initiating events by means of logical AND

and/or OR gates, • Minimal out sets contain those combinations of corn~

ponents, which if they fail, cause system failure.

The probability of occurrence of initiating events can be ob~

tained from generally two sources: (1) historic observation (2) calculations. Historic failure frequencies of all sorts of (mostly mechanical and electrical) components are available in data bases. The occurrence of storms and earthquakes is based on historic observation. The probability of initiating events is evaluated by calculations in-corporating uncertainty of the event parameters.

When creating a fault tree in the design stage, it becomes neces~

sary to investigate all possible failure modes and their mutual relation. The question arises which probability of a TOP~event is ac~

ceptable. As long as human life is not involved, an economic approach to the optimal safety level is most rational.

According to this approach the total generalized cost equals the sum of construction cost, cost of annual normal maintenance and repair during the intended operational life time, and the cost of annual reservation (insurances) to averse unexpected repair and indemnities to third parties. Rational designing requires minimization of total generalized cost, though it is common to include annual reservation for economic losses due to unexpected failure in order to meet with considerations of risk aversion (prestige, reputation).

Incorporating the rates of interest the mathematical expression for the total cost of a breakwater is:

The first term I represents the total investment as a function of the probability of failure, M is the maintenance cost per year also a function of the probability of failure, r the net rate of interest ac~

counting for inflation, D the total damage in case of failure (including repair, damage to other structures, shipping delay, etcetera), and the summation is over N year.

The total cost has to be minimized to find an optimal safety level for the breakwater, or the corresponding probability of failure. In many oases however a minimum failure probability is specified as an

., 17

-objective constraint by the owner, taking into due account also so-cial, moral and political interests and statutorial regulations, as well as considerations about the prestige of the parties responsible for the structure.

The risk analysis of a breakwater is based on the principles of the systems approach. There are two main possibilities of ordening systems: series and parallel systems. For a system of three components in series the functional block diagram and the associated event tree are presented in Figure 3.1.1. The event tree is a logic diagram that gives all possible sequences, starting from an initiating event lead-ing to a consequence for the structure state. The event tree is useful for the analysis of the consequences of an initiating event and it provides a mean to identify TOP~events for a fault tree.

The corresponding fault tree presented in Figure 3.1.1 shows that the failure of one component causes total system failure, symbolised by connecting the TOP~event with component failure by an OR-gate. In the present case three minimal cut sets exists each containing the failure of one component.

The probability of failure of the system is formally written as:

The evaluation of this expression is complex if the events are correlated, which is often the case (viz. Annex I). However, a lower and upper bound can be obtained:

where Pfi represents the probability of failure of system

s

1i and n the number of subsystems. The lower bound is valid if one event con~ tains all others. The upper bound is valid if each event excludes all others. If all events are independent, the probability of the system failure becomes:

In practice the upper bound is a safe and reasonable approxima~

tion for most civil engineering structures,

initiating SYSTEM S11 FAILS TOP .. event SYSTEM S1 FAILS SYSTEM S12 FAILS consequence

l

S₁ fails S1 fails S1 fails S₁ works SYSTEM S13 FAILS

Figure 3.1.1 A three component series system and corresponding event tree and fault tree

The analysis of a parallel system is rather similar. For a system of three parallel components there is only one minimal cut set. It contains all initiating events (Fig. 3.1.2). The probability of failure of the system is expressed by:

The evaluation of this expression encounters the same difficulties as mentioned for the series system, if events are correlated. Here, also a lower and upper bound can be defined:

The lower bound is valid if at least one event excludes one other event. If one event is part of all others the upper bound holds. When all events are independent the expression is:

In most cases the failure bounds for parallel systems can be ex~ pressed as:

initiating TOP-. event SYSTEM S1 FAILS

I

/AND-gate~

consequence

1

S₁ works S₁ works

SYSTEM S11 FAILS SYSTEM S12 FAILS SYSTEM S13 FAILS

Figure 3.1.2b The risk analysis for a parallel system

A more complicated system containing components in parallel and series, is presented in Figure 3.1.3. In this case two minimal cut sets can be distinguished. The first contains only the failure of S_{11 ,} the second S₁₂ and S_{13 •} The expression for the probability of system failure becomes:

An upper bound and lower bound are for each set:

and for the total system:

If all component failure (events) are independent, the result is:

and:

The present discussion reviews static event trees including only functional sequential relations. In general dynamic event trees which cover time dependent aspects should be considered. This approach is possible, but it requires complex methods based on Markov's theory [7], which is not further discussed.

initiating .., 22 «! TOP'"ieVent SYSTEM S1 FAILS

I

OR-. gate

~

sl

fails

sl

works S1 fails S₁ works

SYSTEM Sl1 FAILS component event

SYSTEM S₁₂+S₁₃ FAIL

I

AND""\ gate

~

SYSTEM S12 FAILS SYSTEM S 3 FAILS

Figure 3. 1. 3b The risk analysis of a parallel-series system

3.2. Probabilistic calculations

Traditionally the parameters used in structural design are con~

23

-parameters may vary and their exact value is not known. A probabil~

istic approach accounts for this fact.

The amount of structural strength reserve of a component may be expressed in terms of a reliability function Z, according to:

Z = R - S

where R denotes the resistance of the component against a failure mode and S the actual loading. R and S are expressed in terms of one or more fundamental parameters, which may possess a random character.

The intensity of the loading S is related to the probability of occurrence of the selected initiating event which can be obtained from historic observation and extrapolation techniques. For example, the maximum significant wave height occurring during a certain time span may follow a Weibull distribution.

The resistance R related to the selected failure mode is to be expressed in a mathematical form using physical concepts (structural mechanics/dynamics),

tions are usually

and the fundamental parameters in these formula-stochastic. Their probability distributions are usually considered to be normal (Gaussian) or log~normal, and the variables are treated as

These assumptions simplify validity has to be checked.

independent, unless specified otherwise. the calculations significantly, but the

However, for many failure modes the mathematical~physical

description is not available or rather poor, because the loading and/or the structural behaviour is complex and not fully understood.

The limit state of the considered component occurs at Z=O; the failure state is related to Z<O. The probability of failure is there-fore equal

correspond probability

to the joint probability of parameter combinations which to Z<O. Mathematically this implies integration of the density functions of the parameters involved over the domain of failure. There are several methods with a different level of sophistication to perform the calculations.

The Joint Committee on Structural Safety (CEB, CECM, CIB, FIP, IABSE) distinguishes three levels.

Level I quasi~probabilistic appraoch. Present constructional design methods with relevant safety factors are used. The safety

factor represents for example the ratio of load at failure to admitted working load, as to create a desired space be~

tween characteristic values of strength and working load. Level II Semi-probabilistic approach. Approximation methods are

ap-plied in which normal probability distributions are assumed for both strength and loading, and the reliability function is linearised in a specific point to determine the actual

probability of failure. The following methods are

distinguished:

first order mean value approach The reliability function Z is linearised about the expected mean value of the parameters involved using Tailor~series expansion.

2 first order design-point approach The reliability function Z is linearised about the point of the failure envelope (Z=O) having the highest joint probability density (design~

point). This approach requires an iterative procedure in the case of nonlinear failure envelopes.

3 approximate full-distribution approach Similar as (2) but the exact probability distributions are

equivalent normal distributions in the desi gn-::-poi nt.

approximated by vicinity of the

Level III Full-::-distribution approach. This method accounts for the ex-act joint probability distribution functions including the correlations among the parameters. It usually requires a considerable computational effort.

In the case study presented in this report mainly level II methods are applied. They are suited for design purposes. For impor-tant design calculations a check on the validity and accuracy by means of a level III approach is recommended.

The reliability function is usually expressed in terms of a set of random variables Xi, according to:

For the level II methods the variables Xi must comply with several restrictions. They are to be assumed independent and normally dis~

.. 25 _,

According to the mean value approach the function Z is linearised about the mean value of all variables by Tailor series expansion:

Z ( X

1 , X 2 , ... )

=

Z ( 11 ( X 1 ) , 11 ( X 2 ) , • • • ) + In { ( Xi"' 11 ( Xi )) Z , X .

l

1

From this expression the mean value and standard deviation of Z can be evaluated; the result is:

1/

o <

z ) ..

n:

i Ij { o <

x . ) •

o <

x . ) • z x • z x •

p . .

1 )

2 l J p i , j lJ

where pij denotes mutual correlations among Xi and Xj. In the case of mutually independent variables (p .. •1, p .. •0) this expression becomes:

11 lJ

A measure for the failure probability can be obtained from the corresponding index of reliability S defined by:

S ll( Z)/ o( Z)

If

z

is normally distributed, since normal distributions of X. have

1

been imposed. Thus, the probability of failure follows from the stan-dard normal probability function ~ which is tabulated in mathematical handbooks:

... 26

-3,3 Application to a breakwater

A breakwater represents a rather complex structure with various components,

foundation,

like stones, armour units, crest structure, slopes, toes, filters, etcetera. Consequently, failure trees and event trees are comprehensive,

The failure considered most feasible to illustrate the use of computers for a probabilistic evaluation is the stability of a relevant cross~section. For harbour operation the crest of the break~ water has to remain at design level. The lowering of the original crest by all kind of causes in any cross~section is selected as the TOP-event. This TOP~event is by no means the only important TOP~event. Also, the spread of damage when at one particular position the crest level becomes too low, has to be considered. Moreover, not every sec~ tion is subjected to the same type of wave action. All these aspects have not been elaborated by the group because of time restrictions,

For the present case study the TOP,.. event is CREST TOO LOW. Corresponding potential initiating events are identified as storm~ waves, incident waves, and earthquakes, which are typically stochastic. Gravity is another cause, the effect of which is random because of non..,uniformi ty and inhomogeneity of the breakwater composi-tion and its foundacomposi-tion.

Probabilities of these initiating events based on data acquisi-tion over relatively small periods, are determined by extrapolaacquisi-tion techniques taking into account the random characteristics of the events. Sometimes it is possible to improve the extrapolation by cor-relating a short series of wave data to another phenomenon of which data over a long period is available, for example relating wave data to average water level data [8].

In annex I a general fault tree for rubble mound breakwaters and a fault tree related to a particular breakwater (the case study) are presented, both for the TOP-event CREST TOO LOW.

For the assessment of relevant accident sequences a distiction is made between different categories of subsystems, to wit: geotechnical, hydraulic and structural processes. Each topic has been further evaluated in separate working parties, the activities of which in..,

eluded the calculation of conditional probabilites of accident se-quences, which have been selected as most relevant to the TOP-event. They are listed in Annex I.

The assessment of the absolute probabilities is determined by in-cluding the probabilites of the initiating events reported in the an~ nexes II, III and IV, which comprise the activities of the working parties.

The state of failure of a subsystem or a component is formulated by R<S, where R represents the resistance and S the loading. Resistance and loading usually appear in design formulas, in physical and mathematical models, and in the engineering design approach. It provides a framework to obtain failure probabilities by calculating the limit state of the reliability function Z.

The assessment of the probability of total failure is elaborated using the fault tree. Since many of the system dependencies are not exactly known, a simple approach is adopted assuming independent sys~ terns and components. If possible, some qualitative effects due to ob~ vious dependencies are included in the final results, which is further discussed in chapter 5.

A

lower and upper bound expression of the total risk is used, ac~ carding to:

Max tP(E.))

n 1 :S Min\I n {P(E.)},1} 1

in which E denotes the event of a distinct failure mode (a minimal cut set), which consists of a set of failing subsystems and components that may introduce system failure.

To illustrate the application of a level II approach a simple problem is evaluated, dealing with the stability of armour units under wave loading for which Hudson's formula applies:

with

The block weight is specified by: factor.

p I p -1

s

w

W

=

p gVD3 _{where V is a volume}

s '

Eliminating W yields an expression for the reliability function Z:

z

W

l

(K

- 28 -·

The calculation of the probability of failure applying the mean value approach is presented in Table 3.3.1. The resulting reliability index 8 (Fig. 3.3.1) becomes 1.89 and the corresponding probability

P( Z<O) <!>(-: 1. 89) "' 0. 030.

To improve the accw:acy the linearisation of the Z function is performed not in the mean value, but in the design~point, which is not known beforehand and is calculated iteratively. The results of the design~point approach for the present problem is 8

=

1.92 and hence,

P(Z<O) <!>(-1.92)

=

0.021. The corresponding design~point parameter values are also presented in Table 3.3.1.

8 reliability index a standard deviation

ll mean va 1 ue

Z reliability function

RELI

BtliTV

FUNCTION

zoo

\

\

~EANPOINT

DESIGN POINT

PROBABill TY

/DENSITY

FUNCTION

Figure 3. 3. 1 The mean and design-:point approach

Table 3. 3. 1

parm mean X.

1

Results of level II probability calculations for the stability

formula.

value stand )li

of armour units according to Hudson's

dev desing point

a. z

,Xi a. <z,X. ai)2/a~ value

1 1 1

---~~--~---~~---~-~-~----~---~---~~---·---~---~----D 4.46 0.05 o. 11 75 0,005 4.68 KO 1 6 3.20 o. 7360 o. 21 4 1 3 PS 2.60 0.13 o. 91 8 0.333 2.46 Pw 1.04 o. 02 -0.353 0.049 1. 05 cotga 1. 50 0.05 0.123 0.600 1. 49 H _s 8.00 1. 00 "'11,000 o. 396 9.20

======================================================================

z

3. 00 1. 59

-->

s

1,89

-->

~<..,e> = o.o3o

The linearisation about mean or design-point leads to different results depending on the shape of the reliability function Z, which is schematically presented in Figure 3.3.1 for only one parameter. In most problems there are many parameters involved. In general the design-point method with approximated full-distributions is prefered for nonlinear Z and non-normal distributions.

For the present example the methods do not show a significant difference. In both methods the probability distributions are assumed to be normal. In reality the significant wave height is according to an extreme or Weibull distribution. An approximate full"'\distribution approach may lead to significantly different results.

... 30

-4.

SIMULATION TECHNIQUES

4.1. Computational models

The description of all physical processes around, on and in the rubble mound breakwater with formulas, physical or empirical models, graphs or engineering intuition will always be an approximation of reality. Parameters involved in the mathematical descriptions cannot be measured exactly. For example, the stability factor in Hudson's formula is said to include all unknown effects, such as: the wave period and wave steepness, the possible acceleration forces and the interaction between blocks, the strength and roughness of the blocks, the temperature, erosion and weathering, the storm duration, and the permeability of the structure, Obviously, more understanding of these phenomena is required in order to improve the formulation of the be-. haviour of armour blocks, in particular for slender multileg blocks. The same applies to the stability and behaviour of other components.

This chapter reviews available simulation models at the par-. ticipating institutions in the Netherlands. Some of these models have been used in the calculations of the case study presented in this report. It is emphasized that many other suitable simulation models exist elsewhere.

4.2. Geotechnical models

Flow inside the breakwater

The expertise gained from studies on the coarse granular sill in the large Storm Surge Barrier in the Dutch estuary Eastern Schelde, resulted in well~calibrated numerical models suited to simulate 2..,Dim, turbulent porous flow through inhomogeneous rubble ... mound structures.

As a spin-off a code particularly suited to simulate flow in rubble mound structures due to wave loading, the HADEER code, was completed and calibrated to several large scale tests on breakwater models in the Delta Flume at De Voorts, the Netherlands [6].

- 31 ~

The numerical simulations predict a significant internal set-up due to geometric nonlinearity in the boundary condition and due to air entrainment. This set9up was confirmed by physical model tests, It ac~

tually embodies the influence of low~frequency loading due to internal porous flow generated by water waves. It gives rise to significant consequences for the geotechnical stability of the slopes (Fig. 4.2.1)

. /

/

/ /

CORE

WATERTABLE FLUCTUATION FOR STORM WAVE CONDITIONS

. / / / / PERMEABLE CORE MSL

·-·=1-·-·-HADEER CODE INCLUDES UNSTEADINESS EFFECT VIRTUAL MASS EFFECT TURBULENT FLOW 2DIM

I NHOVOGENE I TY AIR ENTRAINMENT REGULAR WAVES

WATERTABLE FLUCTUATION FOR STORM WAVE CONDITIONS AND A PERMEABLE CORE

Flow in the foundation

If the subsoil on which the breakwater is situated contains a top sand layer, one has to be aware that this layer will respond to wave pressures. Some sands may lose coherence and liquefy. In such a cases the breakwater toe collapses allowing the armour layer to slip down. Next, the unprotected core is open to direct wave attack. This process is a typical example of an accident sequence.

Special laboratory tests on carefully obtained samples are to be performed to evaluate the liquefaction potential in order to determine relevant soil parameters to be input in the overall mechanical stability analysis.

Although liquefaction is a phenomenon that needs more research, some numerical models are suited to approximately determine the li~ quefaction potential and the cyclic generated dissipating excess pore~ pressures, The result of this evaluation is a reduction of the available shear strength of the foundation due to wave action, which can be introduced in a slip failure surface analysis.

Geotechnical stability; slip~failure surface analysis

The deformation at failure state is dependent on the internal friction mobilised in the skeleton formed by a randomly packed rock fill. The internal friction behaviour of rock fill is rather different compared to the one for sands. The intrinsic rock stiffness, the in situ stresses, the surface roughness and erosion show a pronounced effect.

For rock fill a model suited for slip surface stability analysis has to be adapted with a special facility to account for the fric~ tional behaviour. Transient pore pressures generated by water waves and wave impacts, as well as acceleration forces should be input. These pore pressures and accelerations have to be determined by other simulation models beforehand.

Geotechnical stability under wave impacts

In a similar fashion the dynamic reponse due to wave impact at the crest structure on top of the breakwaters has been investigated applying a sophisticated numerical model suited to simulate a dynamic two-phase (pore water and rock fill) medium-structure interaction [5].

'"" 33 ""

This model, the SATURN code, has been verified by physical model tests at different scale (1 :78 and 1 :12) equiped with special pore pressure gauges and 1 inear displacement transducers. The observed be-haviour confirmed the computationally discovered phenomena: significant dynamic pore pressures and significant accelerations.

Consolidation and settlement of the subsoil

I f the subsoil underneath a breakwater contains compressible layers the settlements of the crest will occur in the course of years. A proper study of these settlements is required. The compressibility of the sub~layers must be determined in the laboratory. Various simulation models are available, most of them based on a linear theory (Terzaghi). For a breakwater a non~uniform loading on the foundation is essential.

Calculation procedure

In Figure 4.2.2. the calculation procedure for assessment of the geotechnical stability is presented. It shows how various stimulation models can be used separatedly or successively. For the presented case study the calculations performed have been indicated.

geotechnical data hydraulic data earth.., quake data

l

IT

I

internal flow by waves accelerations impacts

l

wave effects in

l

actual pare-pressures actual

l

!

actual state of stress

sli surfaces settlements

4.3. Hydraulic models

In Figure 4.3.1 a relation scheme is given for the hydraulic loads on a breakwater. The diagram indicates which mathematical models are available or in development.

The actual hydraulic boundary condition due to a storm~wave en-vironment can be determined by simulation models, which calculate the near shore wave climate from the deep sea situation.

Wave hindcast

The code GONO is a numerical wave prediction model in use for the preparation of forecasts as well as hindcasts, It is a hybrid model: the wind sea is described in a parametric way, but swell is treated in a spectral manner. For the wind sea there are two prognostic parameters, the zero~moment wave height and the mean direction. Pure wind~sea spectra are assumed to have a quasi~universal shape: above the spectral peak a f-5 behaviour is assumed, below the peak a linear frequency dependence is taken.

The directional dependence is of the cos2_{a type. Empirical rela~}

tions are used to derive the full set of wind~sea parameters. For the accurate propagation of swell, possibly over large distances, a ray technique is used. Bottom dissipation effects are taken into account, but effects of refraction are disregarded. The model has been applied in regions with depths less than 15 m. The behaviour of the model was studied in quite some detail during the recent Sea Wave Modeling project in a few idealized situations [1 ].

Morphology

The code COMOR is a model with two facilities: initial/final and transient situations. The first predicts initial and long term bottom changes as a result of the construction to be built. Mathematical models for two-dimensional wave fields, for depth and time~averaged flow and for sediment transport are combined into a model for initial bottom changes in front of the construction. The second represents a time dependent morphological model for bottom changes in front of the construction.

overtopping tide

---r---

_ _ _ ...~., _ _ _ "'111 hydraulic boundaries pressures, velocities on breakwater slope

---,

material properties w U'1

mathematical model(s) availab·

mathematical model

in

development·

The code WAMOR predicts bottom changes in large areas, This code uses the subcode WAQUA (time dependent depth averaged flow), the sediment-flow relations, water depth material parameters and the wave influences in the considered area. Initial (after one tide) or time dependent results can be obtained for granular materials. The model can be used for mud and silt where local sediment transport is no longer coupled with local circumstances.

Wave propagation in shallow water and penetration into a harbour The phenomena of refraction and diffraction due respectively to depth or flow velocity variations and lateral transfer of wave energy are included in these models. There are two main types of approach, expressed in the frequency-domain and in the time~domain.

In the frequency-domain a wave potential is used to formulate harmonic linear waves and irrotational motion. In the time-domain a non~linear wave equation is applied. The models are suited for long and steep waves.

The full refraction~diffraction model REFDIF is based on the so-called mild-slope equation [2]. Reflection and transmission at the boundaries are included. At present the model is improved with respect to bottom friction. It will be operational on a supercomputer.

A simplification of REFDIF is the model CREDIZ. The diffraction in the direction of wave propagation is neglected and the boundaries are non-reflective. Bottom friction and wave~breaking can be taken into account

[3].

Ship~induced water motion

To be able to determine the dimensions of a bank protection due to ship~induced water motion, the program SHIWA has been developed.

From the dimensions, the shape and the engine power of a ship, and the dimensions and geometry of a fairway, the speed of the ship can be determined. However, the ship's speed is strongly effected by the ship-induced water motion. In fact, water motion and ship speed are mutually dependent,

The water motion caused by ships can be devided in the following components: primary ship wave, secondary ship wave, and screw-race, The primary wave can be split up into a transversal stern wave, a

37

-slope supply flow and a return current (at the area of water level depression).

The induced water motion imposes forces on banks and bottom of the canal, which can be devided into external and internal forces. Internal forces occur due to the groundwater effects induced by primary and secondary ship waves. These forces play an important role in dimensioning the filter layers.

The direct attacks on the banks by waves referred to as external forces,

toplayers of the protections

[3].

Calculation procedure

which govern

and currents are the design of the

In the case study all these models have not been actually applied. The attention has been focussed on the stability of a cross-section of the breakwater structure. The boundary condition imposed on the section is ofcourse related to the deep sea and near sea wave con~ ditions as well as to the layout of the harbour dams. It is obvious that the optimization of the breakwater position is one of the impor~ tant issues. It has not been included in this study. This restriction was necessary because of limited time and facilities.

4.4. Modelling concrete fracture in armour units

Although phenomena of concrete fracture are well established, no general simulation model that covers the fracture of a placed armour unit due to working load, due to placing method or due to local settlements or to whatever reason, is available, which is mainly at-tributed to the unacquaintedness with the exact position, support and loading. In annex IV a simple model has been applied for the specific case of a armour tetrapod. This model assumes a particular rocking mode critical to the development of extreme tensile stresses leading

~ 38

-4.5. Physical models

Physical model tests have been in use since long. Systematic in-vestigations have sometimes resulted in suitable empirical formulas, of which the Hudson formula is probably the best known example. Many processes are so complex, that the only possibility to investigate their consequences is by the use of physical modelling.

It is usually sufficient to use a small scale model with a scale factor of 10 to 50. Scale effects can be investigated by large scale models, where also the internal flow and the mechanical behaviour of the structure can be measured more accurately. The large Delta flume of the Delft Hydraulic Laboratories can meet these requirements and it is designed for combined hydraulic and geotechnical research.

For quarry stone armour units recently extensive research has been carried out [4]. The investigations resulted into new stability formulas for quarry stone revetments on a relatively impervious core

(the surf similarity parameter is restricted to ~

<

2.5 - 3.5):

z breaking waves nonbreaking waves and cota ~ 3 nonbreaking waves and cota ~ 3

in which Hs/~on

50

is a dimensionless wave height,

s

2

11N

the damage as

function of the (-cota//(Hs/L

0), face profile, N

number of waves, ~ the surf similarity parameter

z

s

₂ the damage level (=A

₂

/o~

₅₀

), A

2 the erosion sur~ the number of waves, L₀ the deep water wave length (=gT2_{/2n) and T the mean wave period.}

z z

The fast evolution of computer development enhanced the use of more and more sophisticated numerical models. Whenever a particular phenomenon can be mathematically formulated, a numerical computational model can be composed. Such a model may have certain advantages above

~ 39

-a physical model, like in scaling simultaneously all processes in~ volved and easily changing conditions and specifications. This makes a numerical model distinct to be applied for probabilistic analysis.

However, many relevant processes cannot be described satisfac~ torily and much will be gained if numerical models are presented as an invaluable complement for the interpretation of physical test results and vise versa.

Quite a number of computer programmes has been developed in recent years to simulate the behaviour of waves and structures. Some of these are described in previous sections. Most hydraulic computer models are used to describe the hydraulic boundary conditions at the location of the breakwater, see Figure 4.3.1.

For geotechnical processes numerical models are available, but physical modelling is a problem because of the stress dependency (deformation, shearing). Centrifuge modelling offers a possibility to cover this deficiency.

No computer programmes are yet available that describe the water velocities and water pressures on the slope of a breakwater or the loads on the armour units. Therefore, it is necessary to simulate the behaviour of a breakwater in a small or large scale physical model.

The hydraulic boundary conditions at the location of the break-water have been used as a starting point for the case study. This means that for the case study no models had to be used in order to calculate these boundary conditions. In reality however it is always necessary to establish the boundary conditions of wave loading from deep water, using various simulation models, flume tests and two-dimensional wave tests.

Three hydraulic failure mechanisms have been considered for a probabilistic approach: wash out of armour, failure of berms and sliding/tilting of a crest structure. Empirical relations obtained from physical experiments have been applied to determine the cor~ responding reliability function.

40

-5.

CASE STUDY.

APPLICATION OF THE RISK ANALYSIS TO A RUBBLE MOUND

BREAKWATER

5.1. Objective of the case study

The case study is given as an example, An existing design for which realistic data are available has been chosen, but much of the required data with regard to the probability distributions of the various parameters has been fancied in a realistic manner.

In the case study the probability of failure in one year has been calculated for the TOP-event chosen as "Wave penetration in harbour too large".

With respect to "wave penetration too large" in the case study it is assumed that this happens when in one cross~section the crest wall settles to 0,5 m below design level. At first sight it seems that this will not cause a too large wave behind the breakwater. However, it should be realised that a great number of such sections exist in a breakwater. They all stand a chance that the crest level drops to 0,5 m below the design level. Also, it is assumed that each failure mechanism considered leads more or less to such a collapse, although probably some will lead to considerably more damage than others, or the damage spreading is quite different. Moreover, it is assumed that once a collapse of 0,5 m of the crest wall has taken place, progres~ sive failure will occur.

5.2. The probability of failure of a rubble mound breakwater

The fault tree

From the general fault tree described in annex I a fault tree for the design of the case study has been composed by leaving out ir~ relevant aspects and rearranging some of the failure mechanisms. This fault tree for the rubble mound breakwater in the case study is presented in annex I as well.

In order to establish the probability of failure of the TOP-event it is required to establish all probabilities of failure of the

subsystems. Some failure mechanisms are a design or construction error for which the probability may be estimated by an experienced designer or contractor. Some failure mechanisms can be described by a simple design formula or they can be studied in a mathematical or physical model, occasionally in combination with other failure mechanisms. For these design formulas and models the probabilistic calculations are outlined in the Annexes II, III and IV.

Failure mechanisms

The results of the probability calculations of the selected failure mechanisms are listed below.

*

Deformation of breakwater too large

Deformation of subsoil too large (Annex II). Calculation of ex~

cessive settlements of compressive subasoil layers below the dam according to the formula of Terzaghi/Keverling Buisman, leads to a probability of failure of 1 x 10- 7•

Collapse of sub~soil cavities (Annex II). Calculation based on engineering judgement leads to a probability of 1 x

10~

7

•

Deformation of core too large. In this study disregarded, This aspect should in principle be considered in conjunction with other settlements.

*

Failure of the crest element

*

~ Base~plate fails due to unfavourable support or slamming (Annex III). When the base~plate is poorly supported on edges crushing of stones may occur so that support surfaces increase. The settlement or cracking of the base~plate can be disregarded. Slamming has been correlated to static wave pressures by physical model tests. And subsequently the required plate strength can be determined. The probability of failure due to slamming has been disregarded assuming that the base-plate is considerably over~ dimensioned with respect to strength.

~ Wave~wall of crest element collapses (Annex IV). The probability of fracture of the wave wall appears to be in the order of 5 x

10~8_•

42 ...

Crest element slides over top of breakwater (Annex Ill). For sliding of the crest wall a simple formula is used and a prob-ability of 12 x 10. 3 has been obtained.

*

Crest element tilts or subsides

Crest element tilts (Annex III). For tilting of the element a simple formula is used assuming that the armour blocks are still in front of the crest wall preventing the wall from a direct wave impact loading. The probability is 13 x 10- 3 •

Overloading of supporting stone underneath the base plate. This aspect has been disregarded since it will stop after some crushing.

Wash-out of material supporting the base~plate. This possibility is negligible based on the assumption that the material is unable to penetrate through the voids of the armour blocks.

Support from inner slope disappears (Annex II). As the crest structure slides backwards, the support capacity of the inner slope becomes insufficient and collapses. A progressive failure occurs. This mechanism is related to crest sliding, which is more critical.

*

Protection from outer slope disappears

Crest wall tilts after armour blocks have disappeared (Annex III). Wave impacts at the crest wall lead to a probability of failure of 26 x 10-3•

*

Instability of armour blocks

Unfavourable support of armour blocks. This aspect has been disregarded.

Fracture of armour during placing (Annex IV). The probability of failure has been estimated to 20 X 10- 3 •

Fracture of armour due to rocking (Annex IV). For rocking leading to ultimate failure assumptions have been made in Annex IV lead.., ing to a probability of failure of 75 x

10~

3

•

Fracture of armour due direct wave action (Annex IV). The prob-ability of failure has been estimated to 1 x

10~

3

•

This mechanism is related to fracture by rocking caused by wave action, which is more critical.

Fracture of armour due to production method, insufficient con~ crete quality and wheatering (Annex IV). The probability of

-3

Hydraulic instability of outer slope (Annex Ill). This aspect is usually covered by the Hudson formula which has been modified for the present case~study. The probability of failure has been cal9 culated as 6 x

10~

3 (secondary armour exposed),

~ Geotechnical instability of outer slope (Annex II). The slip~

failure along a critical slip surface through the dam and loose seabed sands is calculated leading to a probability of 7 x

10~

3

•

*

Support of outer slope disappears

Hydraulic instability of armour at berm (Annex III). From tests a stability factor has been calculated for the berm which can be used in the Hudson formula. The probability of failure for the berm is 0,9 x 10- 3 •

Erosion of bottom in front of breakwater. This aspect has not been evaluated.

Local geotechnical instability of the toe (Annex II). With a slip surface approach the probability of failure has been calculated, taking into account loose seabed sub•layers in which excess pore pressures are generated by repeated wave loading. The result is a probability of failure of 8 x

10~

3

•

This mechanism is related to the total slope failure, which is slightly less critical,

4 Wash~out of inner material below armour. This aspect has been

disregarded because the filters are geometrically stable and par~ ticles of adjacent layers cannot immigrate,

~ Failure due to earthquake action (Annex II). Earthquakes generate excess pore pressures in the seabed sand and cause cyclic ac~ celeration in the breakwater. A corresponding slip surface analysis showed a probability of failure of 23 x 10-3•

Overview of failure mechanisms

Considering all failure mechanisms shown in the fault tree it is obvious that only a few are of real importance, Others have such a low probability that they may be disregarded, if they are independent with respect to other failure mechanisms.

In Table 5.2.1 the results of the combined probabilities are presented leaving out all failure mechanisms with a calculated prob~

~4

ability of less than 10 , and those being in conditional relation to other more critical mechanisms.