World Journal of Mechanics, 2012, 2, 229-238
doi:10.4236/wjm.2012.25028 Published Online October 2012 (http://www.SciRP.org/journal/wjm)
Probabilistic Indicators of Structural Redundancy in
Mechanics
Branko Blagojević1, Kalman Žiha2
1Department of Mechanical Engineering and Naval Architecture, Faculty of Electrical Engineering, Mechanical Engineering and
Naval Architecture (FESB), University of Split, Split, Croatia
2Department of Ship Structures, Faculty of Mechanical Engineering and Naval Architecture (FSB), University of Zagreb, Zagreb,
Croatia
Email: bblag@fesb.hr, kziha@fsb.hr
Received July 20, 2012; revised August 23, 2012; accepted September 2, 2012
ABSTRACT
The paper in the introductory part reviews various definitions and interpretations of structural redundancy in mechanics.
The study focuses on the general structural redundancy of systems after sequences of component failures followed by
possible load redistributions. The second section briefly summarizes the Event Oriented System Analysis and structural
redundancy in terms of the conditional probabilistic entropy. Mechanical responses to adverse loads in this approach are
represented by random operational and failure events in the lifetime. The general redundancy measure in the third sec-
tion of the paper employs the information entropy and goes beyond existing formulations since it includes all functional
modes in service. The paper continues with a summary of traditional redundancy indices. In addition, it proposes an
alternative redundancy index that accounts for the transition to secondary functional level in case of failures of primary
components. The example of a ship structure illustrates the usage of the conditional entropy of subsystems of opera-
tional events and compares it to the traditional and newly proposed redundancy indices. The study at the end investi-
gates how to enhance the safety of structures by using the redundancy based design.
Keywords: Reliability; Event-Oriented System Analysis; Entropy; Redundancy; Redundancy Index; Structures
1. Introduction
Complex load-carrying engineering structures share a
number of properties that relate to the overall structural
safety. The definitions, applications and mathematical
interpretations of important properties such as redun-
dancy, robustness and vulnerability are still vague in dif-
ferent engineering disciplines [1-5]. Although those prop-
erties have been extensively investigated [6-9], the me-
chanical and structural redundancy has been one most
difficult to quantify. It is usually classified into local and
global (overall) redundancy and can be expressed in two
forms, as the system reserve strength and the residual
strength [10,11]. Reserve strength is the margin between
the design load and the ultimate capacity of the overall
structure to sustain the applied load.
The redundancy of complex engineering systems is
commonly related to residual strength that remains in the
structure after one or more components have failed [11,
12]. Descriptive redundancy assessment is characterized
by a number of functional levels and by a number of al-
ternative operational states. Mere deterministic measures,
denoted as structural residual resistance factors and re-
dundant factors which relate the strength of the intact and
damaged structures, are considered as insufficient for
problems of system redundancy.
The structural engineering for the sake of efficiency
often focuses on “fail-safe” or “damage-tolerant” object
that remain operational in the case of the first component
failure. Redundancy in engineering systems in this paper
is assessed as a general probabilistic operational abun-
dance rather than deterministic survival ability in the
case of first component failure. In complex structures
with large number of interacting components possibly in
different uncertain states, the redundancy becomes a sig-
nificant issue in investigation of general system safety.
The study examines the benefits of combining the
probabilistic reliability methods and the event oriented
system analysis of structural redundancy using the same
set of input data. EOSA relates the system redundancy to
the conditional entropy of subsystems of failure events.
Entropy as a measure of information primarily originates
from the information theory [13,14] and lately is gener-
alized as an uncertainty measure in probability theory
[15,16]. The unconditional information entropy has not
been considered as yet as a useful measure of practical
importance for structural systems in engineering.
However, the concept of conditional entropy in the
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA
230
probability theory represents an objective uncertainty
measure for systems of events independent of anything
else but possible events [17]. The redundancy defined by
conditional entropy of failure path is found useful in as-
sessment of lifeline systems efficiency in damaged con-
ditions [18]. Therefore, the paper compares alternative
definitions of the probabilistic redundancy indices to the
entropy-based probabilistic redundancy measure in order
to add impetus to further investigation of system safety
enhancement in mechanical problems.
2. Event Oriented System Analysis (EOSA)
The event-oriented approach considers structures and
structural components as systems and subsystems of
random events in a lifetime. EOSA enumerates all ob-
servable, or at least the most important events, and des-
ignates their status as intact, operational, transitive, fail-
ure or collapsed. To every event status, a probability of
occurrence can be assigned. By grouping events of dif-
ferent status, and associated probabilities, EOSA forms
systems and subsystems of events. Jointly, the event-
oriented system analysis combines modes analysis, enu-
meration of events, probability and information theory in
order to quantify probabilistic systems’ redundancy. Met-
hods such as Secon Moment Methods (SM) First Order
Reliability Methods (FOSM, FORM), Advanced FORM
(AFORM), Second Order Reliability Methods (SORM)
or Monte Carlo simulation and Bayesian methods are on
disposal for probability calculations [19-23]. Most of
these methods have been used in the assessment of safety
of ship structures and structural components. EOSA uses
these methods and provides additional information about
structural behaviour through a calculation of entropies of
systems of events. Additionally, methods of operational
modes and effects analysis such as minimal cut-sets,
minimal tie-sets, event-tree and fault-tree analysis can
identify events [24,25] and determine their relationships.
EOSA methodology is summarized in the sequel.
A system, S, consists of all observable events Ek with
probabilities p(Ek), k = 1, 2, ···, N, where N is the total
number of events. Events can represent different func-
tional states of the structure such as failure or operational
state or some other state of the interest.
Event realization causes that system has a different
functional status “s”. EOSA designates events and sys-
tems’ functional statuses with following indexes: i-intact
(non-damaged), c-collapse, t-transitive, n-non-transitive,
o-operational (with full or reduced capacity), f-failure,
d-damage and combinations. Subsystems of events are
made of the events of the same type. i.e. So is then a des-
ignation of a subsystem of all operational events and Sf
is a subsystem of all failure events. Finite schemes are
commonly used as mathematical presentation of the sys-
tems and subsystems of events [16]:


1
1
N
N
EE
SpE pE



(1)
The systems of events can be also represented as a
summa of subsystems of events:
of
SSS (2)
So ad Sf are subsystems of operational and failure
events


1
1
N
No
oo
o
oo
EE
S
pE pE





1
1
NN
oo
NN
oo
ff
f
ff
EE
S
pE pE







N
f
N
f
o
N = No + Nf is the total number of events in S.
The reliability of a system is equal to the probability of
the occurrence of the subsystem of operational events:

 
1
o
N
o
k
k
RS pSpE

(3)
The probability of failure of a system is equal to the
probability of a failure subsystem:

 
1
of
o
NN
f
f
fk
kN
pS pSpE


(4)
For complete systems is p(Ek) = 1, otherwise p(Ek)
< 1.
Redundant structures are operational even if some of
the structural components have failed. Thus, the redun-
dant systems of events, representing redundant structures,
have to be analysed as functional multi-level operational
systems. A functional level can be understood as a sys-
tem of events comprising all functional states of an ob-
ject. The initial intact state of a structure is modelled as a
system of events on the first functional level (primary
level). After failure of one or more structural components,
the system transits from the first level to the second level
(secondary level). Further failures cause the system to
transit to the third level, and so on.
An event is of status “s”, where l = 1, 2, ···, n is
a level and , and is the number of
events of the same status “s” within a functional level.
ls
k
E
1,2,, ls
kNls
N
EOSA applies the entropy concept from information
theory to assess the effects of the number of events, and
dispersion of their probabilities, as well as the possible
redistribution of loads after failures. The entropy in in-
formation theory is a logarithmic function that measures
uncertainty originally introduced by [13]. The Rényi en-
tropy is a generalization of Shannon entropy, and repre-
sents a family of functionals for quantifying the uncer-
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA 231
tainty of a system. The entropy of a system of events of
order α, for α 0 is [14]:

2
11
1log
1
nn
n
ii
ii
H
Sp




p
(5)
where log base is commonly 2. For the same probabili-
ties of all events, Rényi entropy is

log
n
H
S
n.
Higher values of α give a Rényi entropy which is deter-
mined by events with the highest probability. Lower
values of α give a Rényi entropy that weights events
more equally, regardless of their probabilities. The case α
= 1 gives the Shannon entropy.
Equation (5) represents a measure of uncertainty cor-
responding to either incomplete or complete system of
events. This is convenient for complex systems with large
number of events where probabilities of some events
cannot be determined, or even some events remain un-
known.
System of events can also be considered under the
condition that only the operational events are of interest.
The conditional entropy of system S can be obtained as
follows [26]:

 

1
log
o
o
oo
Nk
o
No
k
pE pE
HSS pS pS

k
o
(6)
The entropy of the operational subsystem does not de-
pend on the probability of a system p(S). Nor does it de-
pend on whether the system is complete or incomplete.
3. The Probabilistic Redundancy Definition
(Redundancy Assessment)
The EOSA relates structural redundancy of the system S
to the uncertainty of the subsystem of operational modes
So [27] by employing the conditional entropy (6), as:

o
o
N
RED SHSS
(7)
The events of a particular interest for redundant struc-
tures are the transitive events Et that cause the system lS
to transit from a current functional level l to a subsequent
level l + 1 denoted . Every transitive event also leads
to the emergence of a new functional state j on the next
level j, of possibly different status such as opera-
tional, failure, collapse, transitive, etc. Functional states
represent distinguished independent ways the object per-
forms its functions with full or with reduced operational
capacity. Newly emerged jth state at level l + 1, after the
transition event k occurred at level l, can be defined
as a subsystem of compound secondary and primary
transitive events
1lS
t
1ls
1ls
S
l
E
lt
j
k
Conditional probability of transition from one level to
the next level is defined as:
SE
.

11lltl lt lt
A practical measure of the probability of system’s re-
sidual strength
1t
pS on the primary level relates on
one hand the probability of damage occurrence
1
pS
with the probability of system collapse , and on
the other, the probability of system operation
1c
pS
1o
pS
in undamaged condition with the probability of the intact
system
1i
pS. The probability of the primary residual
strength
1t
pS equals then to the probability of the
transitive modes and can be expressed as follows:

11 1 11tfco
pSpS pS pSpS
i
(9)
The increase of the probability of residual strength,
1t
pS in (9) practically can involve the diminution of
the probability of the primary intact mode, rather
than of the collapse mode , since
1i
pS

1c
pS
1i
pSc
1
pS for realistic objects. Therefore, the
probability of residual strength for realistic objects has to
be carefully selected in order to ensure the maximal pro-
bability of the primary intact mode, the mi-
1i
pS
nimal probability of the collapse mode and
the
1c
pS
1t
sufficient probability of transitive modes . More-
over, the subsequent functional states also have to satisfy
the safety requirements imposed on a redundant system
after next component failures.
pS
The traditional probabilistic redundancy index RI is
defined as the system’s primary residual strength condi-
tioned by any first component failure 28. Such an index
can be calculated in terms of the probabilities of primary
structural level l = 1 as shown:



1
11
1
t
tf
I
f
pS
RpSSpS

(10)
The complement RR of the probabilistic redundancy
index is defined as the system’s primary collapse, condi-
tioned by any first component failure 29 as shown:

 

11
11
11
11
ct
cf
R
I
ff
pS pS
RpSS R
pS pS
 (11)
The traditional redundancy index RI and its comple-
ment RR do not account for the intact mode.
The probabilistic redundancy factor RF can be defined
30, as the system’s primary residual strength, condi-
tioned by the collapse of the system, as follows:



1
11
11
t
tc I
Fc
I
pS R
RpSS R
pS

(12)
Redundancy in engineering problems in this study is
viewed rather as an operational abundance than mere
survival ability after first component failures. Therefore,
an alternative probabilistic redundancy index proposal is
introduced in the study that considers the system’s pri-
mary residual strength conditioned by operational mode
j
kjk
pSEpSEpE

k
(8)
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA
232
as shown [31,32].


 

11
11
1
1
ti
to
Oo
pS pS
RpSS pS pS

1o
(13)
The primary redundancy indices (10, 11, 12 and 13) do
not account for the redistribution of loads in case of
component failures. The probability of the reserve strength
in case of load redistribution may be more appropriately
expressed by the compound probabilities
21it
j
k
pS E

2i
pS
of the second level intact mode and first level transitive
mode, denoted also as secondary reliability cal-
culated as:
 
1
21
1
t
N
it
jj
j
pS pEpS
2
i
i
(14)
Moreover, the probability of the primary reserve
strength may be viewed as the upper bound on the sec-
ondary reliability. This reasoning can be extended to a
system with any number of functional levels representing
redundant structure. The overall reliability of the multi-
level system that accounts for probabilities of all intact
modes on all levels l = 1, , n is calculated as follows:
  
12ono i in
pSpSpS pSpS (15)
4. Example
The example applies EOSA to investigate the probabilis-
tic redundancy of a longitudinally stiffened structure lo-
cated at the strength deck of a double-hull oil tanker
(Figure 1).
The ship has following general characteristics: length
between perpendiculars 174.8 m; beam 31.4 m; draught
at full load 12.2 m; block coefficient Cb = 0.82; depth
17.5 m; height of neutral axis from bottom 7552 m; dis-
placement (full load draught) 47,400 tons [33].
Structural analysis, done according to the DNV Rules
[34,35], showed that the considered structural configura-
tion remained operational when some of the components
have failed. All longitudinals and plating between longi-
tudinals were involved in redundancy calculation as load
carrying elements giving the total number of seven
structural elements (Figure 1) on the first functional
level (l = 1). After failure of one longitudinal stiffener the
system transits into second functional level with six car-
rying elements on the second functional level (l = 2).
Finally, after failure of two longitudinal stiffeners the
system transits into third functional level (l = 3) with five
remain carrying elements (3 × plating, T girder and one
remaining longitudinal stiffener).
First functional level includes the following random
events: buckling of plating between longitudinals (3 ba-
sic events), torsion buckling of bulb longitudinals (3 ba-
sic events), yielding of bulb longitudinals (3 basic events),
torsion buckling of T girder (1 basic event) and yielding
of T girder (1 basic event). The first functional level has
n = 11 events designatedl
k
A
, k = 1, 2, ···, 11.
Reliability indices are calculated according to [19]:
22
CD
ii
CD
ii
ll
l
i
ll


(16)
where li
C
represent the mean values of stress random
variables. In the cases of yield failure modes the mean
values were taken as 60% of the material yield stress. In
the cases of buckling failure modes the mean values were
taken as calculated from the DNV formulas for critical
buckling stress [35] for the corresponding structural ele-
ment.
D
i
l
represent mean values of load stresses de-
termined in structural analysis.
D
i
l
and
D
i
l
represent
corresponding standard deviations of the variables.
All the stress random variables are assumed to be in-
dependent and uncorrelated, with linear safety margins
for all functional levels. All stress variable are taken to
be log-normally distributed with coefficient of variation
(COV) of 0.7 [36]. Reliabilities, R, and collapse prob-
abilities, pf, are calculated respectively:

11i
ki
RpA
 and

11
1
ci
f
kk
ppA pA
where Φ is the standard normal density function. For
levels with only one state j index is omitted for simplicity.
The mean values of the wave induced bending moments
and design pressure on deck are calculated according to
the DNV Rules. The mean values of the still-water
bending moments were taken as given in the trim and
stability book for the full load state of the considered
z'
y
tt
bt
ht
tb
b3
tp
HP3
HP1HP2
b1b2
2
4
67
5
13
Figure 1. Location and configuration of the investigated
panel of the tanker deck.
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA 233
ship. Statistical properties of random variables for tank-
ers can be found in literature [37-39]. Selected data is
presented in Tables 1 and 2. In this investigation the fol-
lowing elements of the deck geometry were considered
as deterministic variables: thickness of plating tp = 14
mm, span of longitudinals l = 5.08 m, spacing of longitu-
dinals s = 0.8 m, web height (T-profile) ht = 450 mm,
web thickness (T-profile) tt = 14 mm, flange width (T-
profile) bt = 100 mm, flange thickness (T-profile) tb = 14
mm, bulb longitudinals HP 220 × 11.5.
Table 3 shows properties of the tanker deck structure
for all functional (operational) levels.
The number of compound events 1N
lon the first
functional level is: 1N = 2n = 211 = 2048. There is only
one intact functional state represented by the event .
Collapse of either one of the longitudinals HP2 or HP3
(Figure 1) causes transition from the first functional
level to the second functional level. There is 1Nt = 15
transitive events on the first level, denoted , k = 1, 2,
···, 1Nt. The remaining 1Nc = 2032 events on the first level
represent collapse of the structure.
E
1
1
1
i
E
t
k
E
Table 1. Statistical properties of material and loads.
Mean DistributionCOV
Yield stress (mild
shipbuilding steel) 235 N/mm2 Log-normal 0.06
Modulus of elasticity 206,000 N/mm2 Normal 0.01
Stillwater bending
moment (sagging) 296,252 kNm Normal 0.4
Stillwater bending
moment (hogging) 244,690 kNm Normal 0.4
Wave-induced bending
moment (sagging) 1,533,336 kNm Gumbel 0.09
Wave-induced bending
moment (hogging) 1,428,791 kNm Gumbel 0.09
Design pressure (deck) 13.6 kN/m2 Normal 0.09
Table 2. Statistical properties of the deck panel.
Mean DistributionCOV
Width of effective plate flange 800 mm Normal 0.01
Section modulus (longitudinals
and plate flange bi) 326 cm3 Log-normal 0.04
Midship section modulus at deck 16.14 m3 Log-normal 0.04
Table 3. Properties of the deck structure (all levels).
Level
L = 1
Level
L = 2
Level
L = 3
Panel cross sectional area in cm2 392 360 327
Panel neutral axis in cm 9.3 8.9 8.4
Panel moment of inertia in cm4 74,021 71,021 67,785
Events are appropriately grouped according to their
functional state and the structure is modelled by the sys-
tem of events with three subsystems: intact, transitive
and collapse. The system can be written as
1111itc
SSSS.
The probability of intact state on the first functional
level is calculated as:
 
11
11 1
1
1
0.9751
ii i
k
k
pE pSpA
 
The probabilities of transitive events at the first level
can be calculated likewise:
 
611
1111
17
18
0.1853 10
tici
kk
kk
pEpA pApA3



 
711
1111
28
19
0.1853 10
tici
kk
kk
pEpA pApA3



 
9
1111
31011
1
0.5985 10
tici
k
k
pEpA pApA2

The probabilities of the remaining events are:
1
4
t
pE= 0.5985 × 102, = 0.3522 × 107,
1
5
t
pE
1
6
t
pE= 0.1137 × 105, = 0.1137 × 105,
1
7
t
pE
1
8
t
pE= 0.1137 × 105, = 0.1137 × 105,
1
9
t
pE
1
10
t
pE = 0.3673 × 104, = 0.2162 × 109,
1
11
t
pE
1
12
t
pE = 0.2162 × 109, = 0.6981 × 108,
1
13
t
pE
1
14
t
pE = 0.6981 × 108, = 0.1327 × 1011
1
15
t
pE
5
The subsystem of transitive events is modelled as:
 
11 1
12 15
1
11 1
121
ttt
t
tt t
EE E
SpEpEpE




Probability of the subsystem of transitive events re-
presents the probability of transition from the primary to
the secondary level also denoted as the probability of the
primary residual strength and can be calculated as:
 
15
11
1
0.0124
tt
k
k
pS pE

The subsystem of collapse events is modelled as:
 
11
1 2032
1
11
1 2032
cc
c
cc
EE
SpE pE




The probabilities of individual collapse events are not
listed here due to large number of events (Nc = 2032).
Probability of the subsystem of collapse events is:
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA
234
 
2032
11
1
0.0125
cc
k
k
pS pE

One can easily check that on the first functional level
the following relation among probabilities holds:
  
111
0.9751 0.01240.01251
itc
pS pS pS

The second functional level emerges when one of the
longitudinals, HP2 or HP3, collapses. The remaining part
of a structure is still operational, but now with reduced
load-carrying capacity. In this case, it is assumed that
collapsed longitudinal has no remaining load-carrying
capacity.
The load is then redistributed to the remaining ele-
ments on the deck structure and new values of reliability
and probability of failure are calculated.
When any of the longitudinals fails the structural con-
figuration reduces to 6 load-carrying elements: 3 × deck
plating, T longitudinal and remaining two HP longitudi-
nals.
Hence, the system of events on the second functional
level has 9 basic events with probabilities calculated in a
same way as for the first functional level:

2
1
i
pA0.9955, 0.9955,

2
2
i
pA

2
3
i
pA
0.9955,

2
4
i
pA0.9999, 0.9960,

2
5
i
pA

2
6
i
pA
0.9999,

2
7
i
pA0.9999, 0.9855,

2
8
i
pA

2
9
i
pA
0.9855.
Probability of occurrence of one compound intact
event on the second functional level is:
 
9
22
1
1
0.9541
ii
k
k
pE pA

System and subsystems of events, representing deck
structure, are presented on Figure 2. The diagram shows
levels, subsystems and events as well as transition paths
from one functional level to another. This figure illus-
trates complexity of modelling of multi-level operational
redundant structures.
However, a number of random events have extremely
low probabilities of occurrence, which gives the oppor-
tunity to reduce the number of events in future develop-
ment of the EOSA methodology.
Failure of the remaining longitudinal, HP2 or HP3,
causes emergence of the third functional level. Thus, the
second functional level also includes three subsystems:
intact, collapse and transitive. The second level is mod-
elled by the following system of events:
11
2 12121211
11
,,,,,,
tt
it tt
jj
NN
S SSESESES 
c
t
2
89
i
1
This level consist of non-transitive events from the
first level (intact, collapse) and new states that emerge
(21 ) due to occurrence of some tran-
sitive event on the previous level. From 15 transitive
events on the first level 15 functional states will emerge
on the second level. However, only six of those new
states are further transitive (Figure 2). Every transitive
state on level 2 has 3 transitive events causing emergence
of 18 new states on the third level (Figure 2). The prob-
abilities of occurrence of transitive events for the first
functional state j = 1 on the second level are calculated:
21
111515
,,
t
SE SE 
 
6
2222
17
1
5
0.955510
tici
k
k
pEpApApApA


 
8
222
29
1
0.1407 10
tic
k
k
pEpAp A

2
9
c
 
6
2222
378
1
6
0.140810
tici
k
k
pEpApApApA


Probabilities of the remaining transitive events and
collapse events can be calculated in a same way. Since
the total number of compound events on the second level
is 2N = 9713 the probabilities of remaining events are not
listed here. The reliability and the probability
of failure
2i
pS
2c
p
S for the second functional level are:
 
12 1
212 1
11 1
0.0118
ti t
NN N
itit
kk j
kk j
pSpEpEpEp S
 

 
2
i
j
2
c
j
 
12 1
212 1
11 1
4
5.610
tc t
NN N
ctct
kk j
kk j
pSpEp EpEpS



 
The overall reliability of the system with 2 functional
levels,
2o
pS, is probability that transition from the
first level to the second level will result in occurrence of
event representing one of undamaged states of second
level. It includes all probabilities of intact states on the
first level as well as all compound intact states on the
second level of transitive states on the first level (1)
and one intact event on the second level ():
t
k
E
2
1
i
E

212
0.9869
oii
pSpS pS
Adequately, the overall probability of failure is then:

212
4
0.01255.689 100.0131
fcc
pSpS pS


Again,
22
1
of
pS pS
and the second level
system (2S) is complete system of events.
The third functional level (l = 3) arises when failure of
both longitudinals, HP2 and HP3, occurs (not necessarily
simultaneously). On the third level there is 5 structural
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA
Copyright © 2012 SciRes. WJM
235
1S t (subsystem of transitive events, 1Et
k, k= 1 to 15)
1Et1
21S
21S n 1Et1
1Et21Et31Et41Et5
22S23S24S25S
22S n 1Et2
23S n 1Et3
24S n 1Et4
25S n 1Et5
1Et6
26S
26S n 1Et6
27S n1Et7
28S n1Et8
29S n1Et9
210S n1Et10
211S n1Et11
212S n1Et12
213S n1Et13
214S n1Et14
215S n1Et15
1Et7
27S
1Et81Et91Et10 1Et11
28S29S210S211S
1Et12
212S
1Et13 1Et14
213S214S
1Et15
215S
2Et12Et22Et32Et42Et5
32S33S
2Et6
31S35S36S
34S
1Sc1Si
(31S n2Et1)n1Et1
2Et72Et82Et92Et10 2Et11 2Et12 2Et13 2Et142Et15 2Et16 2Et17 2Et18
38S39S
37S311S312S
310S314S315S
313S317S318S
316S
(32S n2Et2)n1Et1
(33S n2Et3)n1Et1
(34S n2Et4)n1Et2
(35S n2Et5)n1Et2
(36S n2Et6)n1Et2
(37S n2Et7)n1Et3
(38S n2Et8)n1Et3
(39S n2Et9)n1Et3
(310S n2Et10)n1Et4
(311S n2Et11)n1Et4
(312S n2Et12)n1Et4
(313S n2Et13)n1Et6
(314S n2Et14)n1Et6
(315S n2Et15)n1Et6
(316S n 2Et16 )n1Et9
(317S n 2Et17 )n1Et9
(318S n 2Et18 )n1Et9
+
+
1S i(subsystem of intact events )
1S c (subsystem of collapse events, 1Ec
k, k = 1 to 2032)
1S (system level 1)
2S (system level 2)
3S (system level 3)
1Sc
1Sc
Figure 2. EOSA model of multi-level system of events representing part of tanker structure.
elements remained to carry redistributed load: 3 × plating,
T girder and longitudinal HP1. This structural configura-
tion represents non-redundant structure, since further
damage of any element will cause the entire structure to
collapse. The number of basic events on the third level is
connected to the number of possible types of failure of
the remaining structural elements, thus giving 7 basic
events .
37n
Probabilities of all intact states are equal:
 
7
33
1
0.9195,1,2, ,18.
ii
jk
k
pS pAj
 
Probabilities of 18 collapse states are:
30.0804,1,2,,18.
c
j
pS j

The third level is modelled as the system of events
consisting of the non-transitive events on the second
level together with the new states on the third level
(compound events). Compound probabilities on the third
level are equal to the probabilities of the transitive events
on the previous levels:
There is 18 operational states on the third level emer-
ging from j = 1, 2, ···, 18 compound transitive events on
the second functional level (Figure 2). Each one of the
18 operational states on the third level has one intact
event. All the remaining events are collapse events, i.e.
there are no transitive events on the third level (non-re-
dundant structure).

3213 21ttt t
j
jk jj
pSEEpSpEpE


 k
The total number of events on the third level is 3N =
12017. The probabilities of 7 basic events are:
  
333
123
0.9874,0.9874, 0.9874
iii
pA pApA
Reliabilities
3i
pS and probabilities of failure
3c
j
pS
are calculated, using the same approach as
before, by means of a computer:
34
1.5977 10
i
pS
 and

35
1.3979 10
c
pS

  
333
456
0.9999, 0.9887, 0.9999
iii
pA pA pA
The third functional level can be also viewed as a sys-
tem of subsystems
333of
SSS, where the subsys-

3
70.9660
i
pA
B. BLAGOJEVIĆ, K. ŽIHA
236
tem of operational states is and

3123oii
SSSS
3123
i
subsystem of failure states is

f
ccc
SSSS.
The overall reliability includes all probabilities of in-
tact states of the first level and the probabilities of transi-
tive and intact states on the second and the third level:
 
3123
0.9870
oiii
pSpS pS pS
The total probability of collapse includes all probabili-
ties of collapse states on the first level as well as com-
pound probabilities of transitive and collapse states on
the second and third level:

3123
0.0130
fccc
pSpS pS pS
Since 0.9870 + 0.0130 = 1, the
system of events that models the structure is complete.

33of
pS pS
The maximum entropy on the first functional level is
log(1N) = log(2048) = 11.0. The conditional entropy of
the first functional level with respect to the operational
modes denoted as redundancy with respect to the opera-
tional mode is according to Equations (5) and (6):
 
111 0.1125 bits.
oo
HSS REDS
The conditional entropy of the system of events for the
second level is then:

2221.0245 bits.
ii
HSS REDS
System redundancy after inclusion of the third level is:
 
333
1.1971 bits.
ii
RED SHS S
Maximum redundancy is 4.1699 bits.

3
max
i
RED S
The first aim of the redundancy based design in the
example was to determine the spacing of deck longitudi-
nal that lead to the most redundant deck topology, that is,
with largest RED(S), starting from the prototype spacing
of 80 cm. The parametric study of structural redundancy
was performed for the range of spacing of deck longitu-
dinal between 63 and 97 cm with following constraints:
The target reliability of a modified structure is equal
or larger than the reliability of the initial structure.
The weight of a modified structure is constant.
The analysis resulted with the highest redundancy
RED(S) = 1.7769 for configuration with b1 = 69 cm, b2 =
80 cm and b3 = 91 cm (Figure 3), which significantly
differs to the prototype (b1 = b2 = b3 = 80 cm). The reli-
ability of that configuration is R = 0.981. The highest
redundancy indicates the structural configuration with
the most uniformly distributed probabilities of operatio-
nal modes for each structural element.
The second aim of the redundancy based design in the
study is to compare the entropy based redundancy meas-
ure RED (S) to the newly proposed probabilistic reliabil-
ity index Ro and to the traditional redundancy index Ri,
Figure 4. The three redundancy measures are computa-
Figure 3. Results of redundancy based design of the part of
ship deck structure.
Figure 4. Redundancy indices Ri and Ro for the first two
functional levels to the entropy based redundancy measure.
tionally not compliant due to different methodologies in
their definitions. Values of Ro indices in the example are
normalized with respect to the minimum gained values of
Ro at each level. The entropy based redundancy measure
RED (S) depends on the probability of operational modes
at all functional levels and the values of redundancy in-
dices Ro and Ri depend on the probabilities of transitional
subsystems on the first level only.
The maximum of the traditional index Ri indicates a
different structural configuration as the best, for b1 = 79
cm, b2 = 80 cm, b3 = 81 cm, on both levels. However, the
maximum of Ro redundancy index shows a good trend as
its values strive towards configuration with maximum
redundancy in each new level. On the first level the best
index value is gained for b1 = 74 cm, b2 = 80 cm, b3 = 86
cm on the second level best index value is gained for b1 =
67 cm, b2 = 80 cm, b3 = 93 cm that almost coincides with
the values of the configuration with highest RED(S).
5. Conclusions
Most of the engineering decisions are governed by con-
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA 237
siderations about matter and energy in the physical space.
Some of the decisions are additionally supported by the
probabilistic and statistical laws governing the probabil-
ity space. However, the event-oriented system analysis
(EOSA) combines the physical and the probabilistic
space to facilitate some of the engineering decisions us-
ing the entropy concept valid in the event space.
The paper compared the entropy-based redundancy
measure and the probabilistic redundancy indices attain-
able within the EOSA. The underlying idea of the study
is that the entropy-based redundancy measure most ap-
propriately captures the intuited meaning of the structural
redundancy even in case of cascades of failures and load
redistribution. The example reveals at this point of the
investigation that the newly proposed redundancy index
Ro in the paper, which relates the primary residual
strength to the operational modes only for the first com-
ponent failure, can be used as an indicator that approxi-
mates structural redundancy. It follows at the primary
level that the trend of the entropy-based redundancy
measure RED(S) that accounts for the multilevel se-
quences of component failures more appropriately of the
traditional redundancy index.
The advantage of the redundancy index, where appli-
cable, over the entropy-based redundancy measure, is in
simplicity and lower calculation effort. However, for
more precise conclusion of redundancy of complex, mul-
tilevel operational structures with a large number of indi-
vidual components, the calculations of all functional lev-
els are necessary. The paper acknowledges that the event-
oriented analysis of redundant objects exposed to succes-
sive component failures, which change the system con-
figuration and provoke a redistribution of demands and
capabilities, is a complicated task, but it also proves that
it is feasible attainable by appropriate software.
Finally, the study confirms that the EOSA has potenti-
alities for structural safety enhancement by adopting the
entropy concept in the maximal redundancy principle for
the redundancy-based design.
REFERENCES
[1] M. Arvidsson and I. Gremyr, “Principles of Robust De-
sign Methodology,” Quality and Reliability Engineering
International, Vol. 24, No. 1, 2008, pp. 24-35.
[2] G. Taguchi, S. Chowdhury and S. Taguchi, “Robust En-
gineering-Learn How to Boost Quality While Reducing
Costs and Time to Market,” McGraw-Hill, New York,
2000.
[3] J. Agarwal, D. I. Blockley and N. J. Woodman, “Vulner-
ability of structural systems,” Structural Safety, Vol. 25,
No. 3, 2003, pp. 263-286.
doi:10.1016/S0167-4730(02)00068-1
[4] A. Eriksson and A. G. Tibert, “Redundant and Force-
Differentiated Systems in Engineering and Nature,” Com-
puter Methods in Applied Mechanics and Engineering,
Vol. 195, No. 41, 2006, pp. 5437-5453.
doi:10.1016/j.cma.2005.11.007
[5] J. W. Baker, M. Schubert and M. H. Faber, “On the As-
sessment of Robustness,” Structural Safety, Vol. 30, No.
3, 2008, pp. 253-267. doi:10.1016/j.strusafe.2006.11.004
[6] B. W. Schafer and P. Bajpai, “Stability Degradation and
Redundancy in Damaged Structures,” Engineering Struc-
tures, Vol. 27, No. 11, 2005, pp. 1642-1651.
doi:10.1016/j.engstruct.2005.05.012
[7] L. Dueñas-Osorio and S. M. Vemuru, “Cascading Fail-
ures in Complex Infrastructure Systems,” Structural Safety,
Vol. 31, No. 2, 2009, pp. 157-167.
doi:10.1016/j.strusafe.2008.06.007
[8] Z. Tian and M. J. Zuo, “Redundancy Allocation for
Multi-State Systems Using Physical Programming and
Genetic Algorithms,” Reliability Engineering and System
Safety, Vol. 91, No. 9, 2006, pp. 1049-1056.
doi:10.1016/j.ress.2005.11.039
[9] J. England, J. Agarwal and D. Blockley, “The Vulnerabil-
ity of Structures to Unforeseen Events,” Computers &
Structures, Vol. 86, No. 10, 2008, pp. 1042-1051.
doi:10.1016/j.compstruc.2007.05.039
[10] S. G. Stiansen, “Interrelation between Design, Inspection
and Redundancy in Marine Structures,” National Acad-
emy Press, Washington DC, 1984.
[11] Y. Feng, “The Theory of Structural Redundancy and Its
Effect on Structural Design,” Computers & Structures,
Vol. 28, No. 1, 1988, pp. 15-24.
doi:10.1016/0045-7949(88)90087-9
[12] K. Chen and S. Zhang, “Semi-Probabilistic Method for
Evaluating Systems Redundancy of Existing Offshore
Structures,” Ocean Engineering, Vol. 23, No. 6, 1996, pp.
455-464. doi:10.1016/0029-8018(95)00051-8
[13] C. E. Shannon and W. Weaver, “The Mathematical The-
ory of Communication,” Urbana University of Illinois
Press, Urbana, 1949.
[14] A. Rényi, “Probability Theory,” North-Holland, Amster-
dam, 1970.
[15] J. Aczel and Z. Daroczy, “On Measures of Information
and Their Characterization,” Academic Press, New York,
1975.
[16] A. I. Khinchin, “Mathematical Foundations of Informa-
tion Theory,” Dover Publications, New York, 1957.
[17] K. Žiha, “Event Oriented System Analysis,” Probabilistic
Engineering Mechanics, Vol. 15, No. 3, 2000, pp. 261-
275. doi:10.1016/S0266-8920(99)00025-9
[18] M. Hoshiya, K. Yamamoto and H. Ohno, “Redundancy
Index of Lifelines for Mitigation Measures against Seis-
mic Risk,” Probabilistic Engineering Mechanics, Vol. 19,
No. 3, 2004, pp. 205-210.
doi:10.1016/j.probengmech.2004.02.003
[19] H. Madsen, S. Krenk and N. C. Lind, “Methods of Struc-
tural Safety,” Prentice Hall, Englewood Cliffs, 1986.
[20] A. H. S. Ang and W. H. Tang, “Probability Concepts in
Engineering,” John Wiley, New York, 2007.
[21] S. K. Choi, R. V. Grandhi and R. A. Canfield, “Reliabil-
Copyright © 2012 SciRes. WJM
B. BLAGOJEVIĆ, K. ŽIHA
Copyright © 2012 SciRes. WJM
238
ity-Based Structural Design,” Springer, New York, 2007.
[22] M. Rausand and A. Høyland, “System Reliability Theory:
Models, Statistical Methods, and Applications,” Wiley-
Interscience, New York, 2003.
[23] R. Y. Rubinstein and D. P. Kroese, “Simulation and the
Monte Carlo Method,” Wiley-Interscience, New York,
2007. doi:10.1002/9780470230381
[24] M. S. Hamada, A. Wilson, C. S. Reese and H. F. Martz,
“Bayesian Reliability,” Springer, New York, 2008.
doi:10.1007/978-0-387-77950-8
[25] D. L. Kreher and D. R. Stinson, “Combinatorial Algo-
rithms: Generation, Enumeration, and Search,” CRC Press,
Boca Raton, 1998.
[26] K. Žiha, “Event-Oriented Analysis of Series Structural
Systems,” Structural Safety, Vol. 23, No. 1, 2001, pp 1-
29. doi:10.1016/S0167-4730(00)00022-9
[27] K. Žiha, “Redundancy and Robustness of Systems of
Events,” Probabilistic Engineering Mechanics, Vol. 15, No.
4, 2000, pp. 347-357.
doi:10.1016/S0266-8920(99)00036-3
[28] C. Palion, M. Shinozuka and Y. N. Chen, “Reliability
Analysis of Offshore Structures,” Marine Structural Re-
liability Symposium, New York, 1987.
[29] R. S. De, A. Karamchandani and C. A. Cornell, “Study of
Redundancy in Near-Ideal Parallel Structural Systems,”
Structural Safety and Reliability, Vol. 2, 1990, pp. 975-
982.
[30] S. Hendawi and D. M. Frangopol, “System Reliability
and Redundancy in Structural Design and Evaluation,”
Structural Safety, Vol. 16, No. 1-2, 1994, pp. 47-71.
doi:10.1016/0167-4730(94)00027-N
[31] K. Žiha, “Redundancy Based Design by Event Oriented
Analysis,” Transactions of FAMENA, Vol. 27, No. 2,
2003, pp. 1-12.
[32] K. Žiha, “Event-Oriented Analysis of Fail-Safe Objects,”
Transactions of FAMENA, Vol. 27, No. 1, 2003, pp. 11-
22.
[33] B. Blagojević and K. Žiha, “On the Assessment of Re-
dundancy of Ship Structural Components,” Proceedings
of the ASME 27th International Conference on Offshore
Mechanics and Arctic Engineering OMAE 2008, Estoril,
15-20 June 2008, pp. 256-262.
[34] N. Veritas, “Ship’s Load and Strength Manual,” Det Nor-
ske Veritas, Oslo, 1995.
[35] N. Veritas, “Rules for Classification of Ships,” Det Nor-
ske Veritas, Oslo, 2007.
[36] I. A. Assakkaf, “Reliability-Design of Panels and Fatigue
Details of Ship Structures,” University of Maryland, Col-
lege Park, 1998.
[37] C. G. Soares and P. M. Dogliani, “Probabilistic Model-
ling of Time-Varying Still-Water Load Effects in Tank-
ers,” Marine Structures, Vol. 13, No. 2, 2000, pp. 129-
143. doi:10.1016/S0951-8339(00)00006-X
[38] K. Atua, I. A. Assakkaf and M. Ayyub, “Statistical Char-
acteristics of Strength and Load Random Variable of Ship
Structures,” Proceedings of the 7th Specialty Conference
on Probabilistic Mechanisms and Structural Reliability,
Worcester, 7-9 August 1996.
[39] J. Parunov, I. Senjanović and C. G. Soares, “Hull-Girder
Reliability of New Generation Oil Tanker,” Marine Struc-
tures, Vol. 20, No. 1-2, 2007, pp. 49-70.