(1)
M. Calabrese and
G. Sorgenti Degli Uberti
Department
of Hydraulic and Environmental Engineering ¡°G. Ippolito¡±
University
of Naples Federico II
Via
Claudio, 21 - 80125 Naples - ITALY - Tel. 0039 081 7683426
Fax:
0039 081 5938936 E-mail: calabres@unina.it,
g.sorgenti@tin.it
Abstract:
In this paper a II level probabilistic analysis of the evolution of beach
nourishments has been proposed. Such an analysis gives an approximate value of
the failure probability with little calculation time. The limit state function
is defined assuming that the nourishment must guarantee an adequate beach
dimension. The analytical formulation proposed by Dean and its extension to
curvilinear contours by Boccotti have been used to evaluate the limit state
function. A comparison of the two formulations has been first performed. The
results have shown that Dean¡¯s model is more severe than the Boccotti¡¯s one
except for high deep-water wave steepness and small ratio between the depth of
the bearing berm of the nourishment and the offshore wave length.
Keywords:
beach nourishment, longshore diffusivity, probability of failure, limit state.
Nourishments are
the current preferred alternative for restoring and stabilising beaches in areas
suffering erosive processes. Such soft structures represent a direct solution to
the sand deficit and if compared with hard structures (e.g. groins, seawalls and
breakwaters) are characterised by limited environmental effects. A nourished
beach generally requires renourishments over time to maintain its function.
Therefore the economic convenience of a beach nourishment is influenced by its
lifetime which depends on the combined action of many factors such as the wave
and tidal climate, the coastal geography, the nature of the fill and native
sediments. The complexity and simultaneity of the effects of these factors
together with the uncertainty of the commonly utilised numerical models make the
probabilistic analysis the most appropriate way to estimate the long-term
performance of a beach nourishment project.
In
this paper a II level probabilistic analysis for predicting the lifetime of a
beach nourishment is proposed. The limit state function has been defined
considering that the main aim of a beach fill is to assure a beach wide enough
to protect coastal areas from damage by storms or to be used for recreational
activities. Its mathematical shape depends on the model utilised for prediction
of beach nourishment project performance. The analytical formulation proposed by
Dean (1992) and its extension to curvilinear contours by Boccotti (1997) have
been considered.
Pelnard Consid¨¨re
(1954) combined the linearized equation of sediment transport and the equation
of continuity, considering the profiles to be displaced without change of form,
to yield:
(1)
in which G is the so-called alongshore diffusivity. The evaluation of the
constant G makes possible to calculate the function x(x,t)
which represents the position of the shoreline at time t, in the section at
distance x from y-axis. In 1992
Dean proposed an analytical expression of G assuming straight and parallel
contours. Boccotti in 1997 discussed a different formulation for G considering
curvilinear and parallel contours landward the bearing berm of the beach
nourishment and straight ones seaward it (Fig. 1).
Fig. 1 Reference system and bearing berm of the nourishment (Boccotti, 1997)
In order to
compare the two formulations, it is necessary to express them as function of the
same quantities. Therefore the utilised formulations are:
(2)
(3)
in which: a0
is the angle between the direction of deep water wave propagation and the
x-axis; kr is the wave number at the water depth, dr, on
the bearing berm; db is the breaking depth; B is the vertical
projection of the dry beach; H0 and c0 are respectively
the deep water wave height and celerity; K is a function of the nature and
dimensions of sediments (Dean, 1992). Moreover:
q0
= a0 ¨C p/2.
In non-dimensional terms, the longshore diffusivity can be expressed as
follows:
(4)
(5)
In order to
obtain general results, independently from the sediments¡¯ characteristics, the
ratio Q
= GB/GD, between the two constants, has been used to
compare them. This ratio allows stressing the range of ao
in which GB assumes negative values GD being generally
positive.
In the analysis,
Q
has been calculated for deep water steepness ranging between 0.02 and 0.10 and
for 0.01 £
dr/L0 £
0.30. The performed analysis has shown that:
(1) for fixed
values of H0/L0 and dr/L0, Q
depends only by ao
and varies symmetrically respect to ao
= 90¡ã. Either GB and GD show an analogous trend;
(2) the
longshore diffusivity estimated by Dean´s formulation is generally major
than that calculated by Boccotti¡¯s model (i.e. Q
<1) and the difference between the two formulations is most evident for small
values of the angle ao.
Only for H0/L0 ³
0.06, dr/L0 £
0.02 and ao
>10o
is Q
>1. In these cases Q
assumes a maximum value for ao
= 30o¸40o.
For normal wave attacks Q
tends to a constant value slightly major than unit;
(3) for dr/L0
£
0.10, GB is negative for ao
minor than a critical value primarily dependent on the ratio dr/L0
while GD remains positive. This means that Boccotti¡¯s model can
predict the reconstruction of a coastal inlet (seaward shoreline convexity)
while Dean does not.
Results for the
extreme values of the examined ranges are shown in figures 2 and 3.
Fig. 2 Q = GB/GD for H0/L0 = 0.02 Fig. 3 Q = GB/GD for H0/L0 = 0.10
The analysis has
therefore stressed that generally the choice of the Dean¡¯s formula to estimate
the longshore diffusivity, makes to predict a more severe erosion in comparison
with the Boccotti¡¯s model.
In order to take
into account the randomness of the wave characteristics, a II level
probabilistic analysis of a rectangular nourishment has been proposed.
With reference
to a rectangular nourishment with a longshore length b and a cross-shore
dimension a, the solution of equation (1) is:
(6)
where:
and
As the usual aim
of a beach fill is to assure a beach wide enough to protect coastal areas from
damage by storms or to be used for recreational activities, the limit state
function has been defined as:
(7)
xlim
being the
minimum dimension to guarantee to the beach.
The reliability
of the nourishment is expressed by the following formula:
(8)
where P(g£0)
is the probability that the limit state function assumes a non positive value
(probability of failure).
The basic random
variables H0, T0, ao,
which appear in
the expression of G, whose distribution has been evaluated by actual wave field
data collected at Ponza station, have been transformed into a set of independent
standard normally distributed variables by a Nataf transformation (Ditlevsen and
Madsen, 1996). The failure surface has been linearized using a Taylor expansion
series around the design point. The probability of failure has been approximated
by:
(9)
where F
is the standard normal distribution function and b
is the first order reliability index, which is defined as the smallest distance
from the origin to the failure function, in the space of transformed variables.
The
probabilistic analysis has been performed considering the two analytical models
proposed by Dean and Boccotti. The ratio between the probability of failure
estimated by Boccotti´s model, PfB, and that obtained with
Dean¡¯s formula, PfD, is shown in fig. 4 for mean offshore wave
direction ao=60o
and 90o and x=b/8 and 3b/8.
Fig. 4 Ratio PfB/PfD vs time, for fixed value of the mean direction of wave propagation at different sections of the nourishment.
In the analysed
cases, the Boccotti´s model has given, for a fixed time t, a probability
of failure minor than that obtained with Dean´s formulation, in agreement
with the deterministic analysis. The probability of failure assumes the smallest
values for the section near the origin of the reference system, independently on
the mean value of the direction of wave propagation.
The performed
analysis has shown that the longshore diffusivity proposed by Boccotti, assuming
curvilinear and parallel contours landward the depth of closure of the beach
nourishment and straight ones seaward it, is generally minor than that
calculated by the Dean¡¯s model which consider straight and parallel contours.
The difference between the two models is more evident for oblique wave attacks,
tending to diminish for normal wave attack.
The longshore
diffusivity estimated by Boccotti¡¯s formula, for dr/L0 major then 0.10, is negative for ao
minor than a critical value principally varying with the ratio dr/L0.
This means that unlike Dean¡¯s model the Boccotti¡¯s one is able to predict
the reconstruction of a beach inlet.
Only for dr/L0
< 0.02 and Ho/Lo
³
0.06 the Boccotti¡¯s model predicts a more severe erosion of the beach
nourishment.
In order to take
into account the randomness of the wave characteristics, a II level
probabilistic analysis has been applied to estimate the reliability of a
rectangular nourishment. The proposed First order Reliability Analysis has the
advantage of furnishing an approximate evaluation of nourishment¡¯s hazard with
little calculation time. The analysis has shown that also in this case the two
considered analytical models give different probability of failure. Only a
comparison of both model results with field data can show their capability to
predict beach nourishment performance and can give informations on model
uncertainties to introduce in the probabilistic analysis.
A more precise
estimation of the nourishment lifetime, for a fixed value of the probability of
failure, can be performed by a III level probabilistic analysis, in which is
possible to take into account the real statistical distributions of the random
wave characteristics.
References
BOCCOTTI P. (1997): Idraulica Marittima. Ed. UTET ¨C Torino. Italy.
DEAN R.G., (1992): Beach nourishment: Design Principles. Proc. Short Course on Design and Reliability of Coastal structures attached to the 23th Int. Conf. Coastal Engineering. Venice, Italy.
DITLEVSEN O. & MADSEN H.O. (1996): Structural Reliability Methods. John Wiley and Sons, Chichester, West Sussex, England.
PELNARD CONSIDERE (1954): Essai de th¨¨orie de l¡¯¨¨volution des formes de rivage en plages de stable et de galets. Quatri¨¨me Journ¨¨es de l¡¯Hydrauliqe, Les Energies de la Mer, Question 3.