Optimal Engineering Design Method that Combines Safety
Factors and Failure Probabilities: Application
to Rubble-Mound Breakwaters
Enrique Castillo
1
; Miguel A. Losada
2
; Roberto Mı
´
nguez
3
; Carmen Castillo
4
; and Asuncio
´
n Baquerizo
5
Abstract: This paper presents a new method for engineering design that allows controlling safety factors and failure probabilities with
respect to different modes of failure. Since failure probabilities are very sensitive to tail assumptions, and safety factors can be insufficient,
a double check for the safety of the engineering structure is done. The dual method uses an iterative process that consists of repeating a
sequence of three steps: 1 an optimal in the sense of optimizing an objective function classical design, based on given safety factors,
is done, 2 failure probabilities or bounds of all failure modes are calculated, and 3 safety factor bounds are adjusted. The three steps
are repeated until convergence, i.e., until the safety factor lower bounds and the mode failure probability upper bounds are satisfied. In
addition, a sensitivity analysis of the cost and reliability indices to the data parameters is done. The proposed method is illustrated by its
application to the design of a rubble-mound breakwater.
DOI: 10.1061/ASCE0733-950X2004130:277
CE Database subject headings: Failure modes; Nonlinear programming; Optimization; Probabilistic models; Reliability analysis;
Safety factors; Sensitivity analysis; Rubble-mound breakwaters.
Introduction
The phases that an engineering structure undergoes are construc-
tion, useful life, maintenance and repair, dismantling, etc. Each
phase has a duration of time associated with it. During each of
these phases, the structure and the environment experience a con-
tinuous sequence of outcomes that have to be analyzed during the
project Ditlevsen 1997. The objective of project design is to
verify that the structure fulfills the project requirements during
these phases. Initially, the engineer must decide the duration of
the useful life of the work being designed. This duration deter-
mines the resulting design.
Next, the modes of failure of the structure must be defined. A
mode describes the form or mechanism in which the failure of
part of the structure or one of its elements is produced. Each
mode of failure is defined by its corresponding verification non-
failure equation that admits different representations as, for ex-
ample:
g
i
* x
1
, x
2
,..., x
n
=
h
si
x
1
, x
2
,..., x
n
h
fi
x
1
, x
2
,..., x
n
-1 0 (1)
where h
si
( x
1
, x
2
,..., x
n
) and h
fi
( x
1
, x
2
,..., x
n
) =two opposing
magnitudes stabilizing to overturning forces, strengths to ulti-
mate stresses, etc. that avoid and produce the associated mode of
failure, respectively, i refers to the mode of failure, and
( x
1
, x
2
,..., x
n
) =values of the variables involved. Checking
whether or not this equation is satisfied, the safety of the structure
with respect to such a mode of failure can be determined. If Eq.
1 holds, failure does not occur; otherwise it does.
This check can be done from two different points of view,
denoted here as 1 classic or deterministic, and 2 probability
based.
In the former, and since engineering design cannot always
strictly be safe, verification equations cannot be used for design.
So, they are modified to increase safety and this leads to the
safety constraint:
h
si
x
1
d
, x
2
d
,..., x
n
d
h
fi
x
1
d
, x
2
d
,..., x
n
d
-F 0; F 1
where x
1
d
, x
2
d
,..., x
n
d
=design values of the variables
( X
1
, X
2
,..., X
n
), and F=safety factor associated with the mode of
failure. Thus, in a classic design the design equations or con-
straints are written in terms of safety factors.
In the case of climatic actions, x
1
d
, x
2
d
,..., x
n
d
can be obtained
from the state variables, H
s
and T
¯
z
. Definition of these state
variables requires a stochastic model.
Safety factors have the advantage of being easily interpreted in
terms of their physical or engineering meaning, but do not give
clear information on the reliability of the structure.
1
Professor, Dept. of Applied Mathematics and Computational
Sciences, Univ. of Cantabria, Avda. Castros s/n, 39005 Santander, Spain.
E-mail: castie@unican.es
2
Professor, Grupo de Puertos y Costas, CEAMA, Univ. of Granada,
Avda. del Mediterra ´neo s/n, 18071 Granada, Spain.
3
Assistant Professor, Dept. of Applied Mathematics, Univ. of
Castilla–La Mancha, 13071 Ciudad Real, Spain.
4
Assistant Professor, Grupo de Puertos y Costas, CEAMA, Univ. of
Granada, Avda. del Mediterra ´neo s/n, 18071 Granada, Spain.
5
Assistant Professor, Grupo de Puertos y Costas, CEAMA, Univ. of
Granada, Avda. del Mediterra ´neo s/n, 18071 Granada, Spain.
Note. Discussion open until August 1, 2004. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and pos-
sible publication on January 22, 2002; approved on July 7, 2003. This
paper is part of the Journal of Waterway, Port, Coastal, and Ocean
Engineering, Vol. 130, No. 2, March 1, 2004. ©ASCE, ISSN 0733-
950X/2004/2-77– 88/$18.00.
JOURNAL OF WATERWAY, PORT, COASTALAND OCEAN ENGINEERING © ASCE / MARCH/APRIL 2004 / 77