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Ŕ periodica polytechnica

Civil Engineering

52/2 (2008) 83–89

doi: 10.3311/pp.ci.2008-2.04

web: http:// www.pp.bme.hu/ci

c

Periodica Polytechnica 2008

RESEARCH ARTICLE

Hydraulic failure probability of a dike

cross section

László Nagy

Received 2008-03-08, accepted 2008-10-21

Abstract

This paper is a brief introduction on the determination of

separate ﬂood plain basins, the selection and determination of

characteristic ﬂood stages inducing typical economic impacts,

and the principles of taking the safety factor or the probability

of failure of the ﬂood defences into consideration in ﬂood risk

mapping. The failure probability is the origin from the vari-

ability of the soil physical parameters and from the constantly

changing water level.

Keywords

Probability of failure ·conventional safety factor ·ﬂood risk ·

dike breach ·soil characteristics ·hydraulic failure

László Nagy

Geotechnical Department, BME, M˝uegyetem rkp. 3. Budapest, H-1521, Hun-

gary

1 Introduction

Flood risk mapping is a cartographical representation of ﬂood

and ﬂood damage characteristics of diﬀerent probability. The

maps are basic tools in ﬂood prone areas for land use planning,

for priority setting in the ﬁeld of investments for the establish-

ment or improvement of ﬂood security, and they are also essen-

tial for insurance planning and for increasing the public aware-

ness of risk [14,17,18,20].

The important characteristics of ﬂoods inﬂuencing possible

damages are the expected water level (or the expected depth of

ﬂooding), the frequency or return period of diﬀerent water lev-

els, ﬂow velocity conditions, and ﬂood duration. All of these

characteristics can be represented in a ﬂood risk map.

Flood risk maps are usually compiled for unprotected ﬂood-

plains of river or creek valleys. In such cases the surface of the

water ﬂowing in the river bed can be computed as a variable

unsteady ﬂow in an open channel. Diﬀerent water surfaces cor-

responding to discharges of diﬀerent probability are determined,

and the horizontal projection of the respective water levels to the

terrain indicate the limits of ﬂood of diﬀerent probability. Char-

acteristic depths of ﬂooding are easy to derive from detailed to-

pographic maps or digital terrain models. Such ﬂood risk maps

are usually used for land zoning or for the planning of structural

ﬂood alleviation schemes.

In Hungary, where 97 % of the ﬂood plains are already pro-

tected, we believe that the risk of damages can also be related to

the stability or safety of the ﬂood defence structures, dikes, and

conﬁnement dikes. The length of the Hungarian ﬂood dikes is

more than 4200 km, so the ﬂood risk is primarily a factor of the

stability of the dikes.

2 The inconsistency of soil characteristics

The data or research ﬁndings that support the calculation of

the degree of safety from the parameter of shear strength or the

coeﬃcient of permeability are normally scarce. It is common

practice to calculate the central factor of safety from the aver-

age of research ﬁndings. A designer whose calculation takes

into account the smallest of the available measurement results

against the most unfavourable combination of loads exercises

Hydraulic failure probability of a dike cross section 832008 52 2

utmost caution. This calculation yields lower resistance values

than the degree of safety calculated from averages. If a system

still complied with the required degree of safety, the designed

structure must have been uneconomically large. That has led

on to a paradoxical situation: as spending on exploration grew,

more and more studies were performed and the likelihood of

receiving poorer and poorer resistance values kept increasing

along with the safety of designing, which in turn kept driving

the cost of construction higher.

Several researchers have studied soil characteristics as statis-

tical values, such as the distribution and variability factor of soil

features (Table 1). However, a review of the literature failed to

identify data concerning studies on the coeﬃcient of permeabil-

ity and the type of distribution.

As a material used for supporting loads and for construction,

soil is a substance that exhibits utmost variation in homogeneity.

While a ten percent coeﬃcient of variation (Cv=10%) repre-

sents poor quality for concrete, the value of Cv=0.4 should be

viewed as satisfactory with some soil characteristics (see Fig. 1).

Fig. 2 shows the results of 54 studies concerning the angle

of internal friction and 91 studies of shear strength of the ex-

plored sandy and rich clay soils in the ﬂood area of Köröszug.

The results clearly demonstrate the relatively low coeﬃcient of

variation for sand.

3 Safety of ﬂood dikes

The ﬂoods after 1945 have caused 140 embankment failures,

of which 83 (58%) were due to overtopping (52 during the 1956

icejam ﬂood on the Danube), 23 (16 %) to hydraulic soil failure,

10 (7 %) to saturation and 2 (1,5%) to leakage along structures,

other identiﬁed 11 (7,5%), while no cause could be identiﬁed

positively in the case of 14 (10%) [15, 16, 18, 19]. In the pro-

tected ﬂood plain basins the occurrence of the various loss types

can be related to the ﬂood stages aﬀecting the stability or safety

of the ﬂood defences. The total obtained is 143 instead of 140

due to the fact that in three cases diﬀerent mechanisms of failure

were named, which could not be judged as to their correctness.

Evidently, the completeness of the list cannot be guaranteed.

Improvements over the past 150 years involved but rarely any

change in the original trace of the embankments. Explorations

of the subsoil and soil mechanical tests have been introduced as

late as 3540 years ago, which recently revealed that the original

trace passes over areas with adverse soil conditions, where the

soil proﬁle contains:

– the meander crossings with its diﬀerent soil layers,

– layers of organic soil or peat,

– dispersive soils,

– loose, poorly graded ﬁne sands in the vicinity of the surface,

etc.

The programme for the investigation of 4200 km ﬂood dikes

was compiled in the 1980s for exploring the subsoil of ﬂood

embankments and for identifying the potential sections of piping

failure. The basic considerations underlying the method are as

follows:

– the subsoil under long embankments of moderate height must

be investigated,

– the soil proﬁle must be explored continuously (virtually by

metres), and

– the subsoil consists generally of a cohesive cover over layers

becoming increasingly coarser with depth.

3

17

22

15

12

10

7

3

11

0

5

10

15

20

25

15 16 17 18 19 20 21 22 25 26

Heavy clay's inner friction angle

Number of samples

Fig. 2. The angle of internal friction of soils explored in the ﬂood area of

Köröszug

In order to carry out the investigation on the stability of the

dikes, the study must be divided into characteristic sections,

within which the following should be presumed more or less

constant:

– the high of the crest,

– the stratiﬁcation of foundation soil and the quality of the lay-

ers,

– material of the existing dike as well as that of the reinforce-

ment or new defences,

– typical cross-section of the existing dikes, and

– phenomena observed along the dikes during ﬂoods.

The section conforming to the characteristics of the founda-

tion soil has a special importance and needs special care. In

the course of the investigation the safety of the embankments

Per. Pol. Civil Eng.84 László Nagy

Tab. 1. The distribution and variability factor of soil features

Soil properties Distribution Coefﬁcient of variation

Normal Lognormal Other

Water content

66 % Corotis [4] 33 % Corotis [4] Pearson IV or VII 0,15-0,19 Rétháti [23]

Davidson [5] Rétháti [23] 0,02-0,2 Borus-Rév [1]

Holtan [8]

Morse [13]

Wet density

Brust [3] 0,011-0,028 Borus-Rév [1]

Ike [9] 0,03-0,05 Evangelista [6]

Prince [21]

Rourke [?]

Particle density Shultze (1971)

Void ratio 80 % Shultze [24]

Saturation Rétháti [23]

Liquid limit

80 % Shultze [24] 33 % Corotis [4] Rétháti [23] 0,11-0,38 Rétháti [22]

66 % Corotis [4]

Lumb [11]

Plasticity limit Lumb [11] Corotis [4] 0,04-0,10 Borus-Rév [1]

Plasticity index Lumb [11] Rétháti [23] 0,26-0,54 Rétháti [23]

Shear test

Hooper [9] 0,15-0,31 Morse [13]

Insley [10] 0,17 Weber [26]

Wu [27] 0,05-0,14 Schultze [24]

Friction angle 50 % Shultze [24] 0,06-0,11 Harr [7]

Cohesion Lumb [11] 0,42 Weber [26]

0,26-0,68 Lumb [12]

Fig. 1. Coeﬃcient of variation of soil characteris-

tics

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

concrete water content wet density folyási határ shear test cohesion friction angle perme. coef.

Coefficient of variation

against piping failure is determined by successive approxima-

tions involving several disciplines, like geophysics, hydraulics,

soil mechanics, and surveying. To determine the longitudinal

proﬁle of the long dikes and the individual sections, one of the

best methods is the permanent horizontal geo-electric probing

with a 1,0 meter electrode distance. The application of this

method makes the exploration of continuous stratiﬁcation pos-

sible. This method also reduces the cost of exploration, while

the application of more expensive methods may be required less

often and only for the identiﬁcation of the layers at easily deter-

minable points.

4 Determination of the conventional safety factor

Controlling the safety factor of the embankment divided into

characteristic sections must be accomplished section by section,

according to standard methods speciﬁed in appropriate guide-

lines and standards. The conventional safety factor is:

n=R/Q(1)

where Ris the resistance (or strength), and Qis the action ef-

fect (load). Using and transforming the equations determine the

safety factor of the defences at actual water stages, and the ﬂood

levels corresponding to previously selected safety factors can be

determined.

Hydraulic failure probability of a dike cross section 852008 52 2

Fig. 3. Failure probability at diﬀerent water stages

Fig. 4. Better and worse dike

Per. Pol. Civil Eng.86 László Nagy

Fig. 5. The occurrence probability of a failure of a dike

So we have the opportunity of deﬁning the ﬂood hydrograph

peaking at the level corresponding to the loading capacity of the

defence structure. Since the most vulnerable cross sections of

the defences are also known, the ﬂood hydrographs represent-

ing the loading capacity are to be transformed to these possible

breach points. The loading capacity of the defences can be deter-

mined by repeating the computations carried out earlier in order

to deﬁne the extension of the ﬂoodplain of 1 % probability of

inundation, the extension of the ﬂood plain section threatened

by the stage corresponding to [25].

Advanced dimensioning methods consider both the impacts

inducing (Q) or hindering (R) the breach to be independent and

probabilistic variables. It is obvious that from the viewpoint of

stability all the combinations of load and resistance are disad-

vantageous where R<Q, represented in the ﬁgure with the

barred territory. The size of this territory is equal with the fail-

ure probability and therefore is appropriate for characterizing

the magnitude of risk of the given section.

5 The probability of failure at ﬂood dikes

In ﬂood protection dikes both load and resistance develop

along certain probabilities. Load is interpreted in terms of the

probability of water levels. The variation of soils and soil char-

acteristics prevents us from identifying in other than probabilis-

tic terms what resistance to failure a ﬂood protection dike will

have under certain water level loads (probable water levels).

When calculating the probability of failure, Q(w) is used to rep-

resent the load probability function, as it is the function of water

levels, whilst R (w) stands for the probability of resistance func-

tion, as it has been calculated from water levels.

The relation between load and resistance may be expressed

by the safety margin (SM):

SM =R(w) −Q(w), (2)

which is also a probabilistic variable. The failure probability

expresses the probability of the opportunity of load exceeding

resistance

pf=P(Q>R)) (3)

or

pf=P(SM ≤0)(4)

The failure probability can be determined either from the avail-

able soil physical data, applying probabilistic design methods

for the whole calculation system or from the traditionally cal-

culated safety factors using a semi-deterministic approach. For

ﬂood dikes the value of failure probability generally must be:

pf<0,01,(5)

Hydraulic failure probability of a dike cross section 872008 52 2

Fig. 6. The hydraulic failure probability of an old

and a developed dike

80.00

81.00

82.00

83.00

84.00

85.00

86.00

87.00

88.00

89.00

0.0001 0.0010 0.0100 0.1000

probability

elevation (m)

water st age

old dike

developed dike

P

f

new

= 0,41% P

f

old

= 2,1%

DEVELOPMENT

which means that among all possible combinations of load and

resistance values only 1 % would lead to breach. In other words,

in 1 % of possible cases will be Q(w) > R(w).

I have prepared a detailed calculation to evaluate the safety

of the protected ﬂood areas along the Upper Tisza and the Sajó

rivers. Based on the calculations and the dike failures of the past

35 years, it is recommended to provide the

pf<10−3(6)

probability of failure of a cross section. At the present stage

of the research, it can be identiﬁed as a boundary value for the

probability of inundation (the probability of failure of a ﬂood

control dike multiplied by the probability of a ﬂood event)

pf<10−5.(7)

The calculations suggest that these values may also be applied in

safety mapping. At present, there is no requirement in Hungary

that speciﬁes an acceptable value for the probability of failure.

It would only be proper to ask why would we use failure prob-

ability instead of the safety factor that we became accustomed

to in practice? The answer is:

•we can characterize the system of defence structures,

•we can obtain the reliability of our results (uncertainties can

be handled), and

•evaluation of risk is possible.

The hydraulic failure probability of a dike with conventional

geotechnical methods can be caluclated for a given water stage.

Repeating the calculation for more water stages gives the fail-

ure probability as a function of the height. Fig. 3 represents the

results of the calculated values of the failure probability in the

possible range of water stages, in addition to the probability of

occurrence of water stages in case of a given proﬁle of a dike

[17]. Diﬀerent failure probabilities are depicted on Fig. 4. de-

pending on the water level. Since the failure probability and

the occurrence of water stages are independent, the probabil-

ity of their joint occurrence can be calculated as the product of

the multiplication of their probability, that is R(water level) ·Q

(water level).

Naturally, we are only aware of the size of resistance (R) and

size of load (Q) functions to a certain level of probability as both

are probability variables (Fig. 4).

Investigating the R(w) ·Q(w) function, the occurrence prob-

ability of a failure of a dike proﬁle can be characterised by the

maximum value of R(water level) ·Q(water level) function.

This consideration is interesting enough for further investiga-

tion.

The hydraulic failure probability of a dike at a certain water

stage is shown in Fig. 6. After the proposed development the

new dike failure probability is less then twenty % of the old one.

6 Conclusions

How safe is any given dike? The answer is provided by a

probabilistic risk assessment, the beneﬁts of which were de-

scribed along with a standard for tolerable risk. It was stressed

that in the absence of analytical techniques, the diﬃculty of as-

signing probabilities can be addressed through the use of experi-

enced engineering judgement who is familiar with the dike and

with all investigations and previous studies at their disposal. It

was proposed that a risk could become a systematic and com-

prehensive framework for the application of engineering judge-

ment.

Risk is the product of failure probability and consequences of

the failure. The application of failure probability in the evalua-

tion of existing and also in design of new ﬂood defence struc-

tures gives us the possibility of adapting these problems to the

risk standards. A standard for tolerable risk is needed in con-

junction with a risk analysis to evaluate dam safety, its purpose

being to permit decisions on dike safety remedial work to be

based directly on risk in a consistent and quantiﬁable manner.

Per. Pol. Civil Eng.88 László Nagy

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