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Y. Murotsu
Professor,
Department o f Aeronautical E ngineering,
Mem. ASME
M. Kishi
Research Associate,
Department
o f
Naval A rchitecture.
H. Okada
Professor,
Department o f Naval Architecture.
Y. Ikeda
Lecturer,
Department
o f
Naval Architecture.
S. Matsuzaki
Graduate S tudent,
Department o f Aeronautical Eng ineering.
University
o f
Osaka Prefecture,
Sakai, Osaka, Japan
P robabilistic Collapse A na lysis
of Offshore Structure
This paper proposes a method for probabilistic collapse analysis of an
offshore
structure. Wave loads are estimated by using Stokes third-order theory and Mori-
son s formula. Plastic collapsing is evaluated by taking a ccount of the combined
load effect to generate the safety margins, using a matrix method. Probabilistically
dominant collapse modes are selected through a branch-and-bound method. The
proposed
method is
successfully
ap plied to
a
jacket-type
offshore
platform.
Introduction
Many studies have been made
of
reliability analysis
of
offshore structures
[1-13].
Some have been concerned with
the failure of the structure caused by an extremely high wave
[1, 3, 7, 9], while the others have treated the problems related
to wave-induced dynamic motions
[5, 6, 8]. At
first,
an
offshore structure was modeled for reliability analysis as a
weakest link system [1] or a few failure modes were specified
[3,
5], which
did not
take due account
of
the system redun
dancy. Then, some approaches were proposed for evaluating
the reliability
of the
complete system
[2, 9, 10, 12, 13].
However, there remain many works to be done for a large
structure [11, 14] which has too many failure modes to
identify all of them and to estimate its reliability.
This paper is concerned with probabilistic collapse analysis
of an offshore structure caused
by
extreme wave loading.
A
finite amplitude water wave approximated by Stokes third-
order theory
is the
input
to the
offshore structu re,
and the
resulting wave loads
are
calculated
by
using Morison's for
mula. The wave height and Morison's force coefficients are
treated as random variables. The first-order-second-moment
method
is
used
to
approximate
the
stochastic properties
of
the wave-induced loads. Structural failure is defined as pro
duction of large nodal displacement due to plastic collapse
in
the structure. The interaction of the bending moment and
Contributed by the OMAE Division and presented
at
the 4th International
Symposium
on
Offshore Mechanics
and
Arctic Engineering, ETCE, Dallas,
Texas, February 17-22, 1985, of TH E
AMERICAN SOCIETY
OF MECHANICAL
ENGINEERS.
Manuscript received by the OMAE Division, July 2, 1984; revised
manuscript received September 4, 1986.
axial force upon
the
plasticity condition
of
the element
is
taken into account. A matrix method is applied to generate
the failure modes and their mode equations. The stochasti
cally dominant failure paths, i.e., sequences
of
plastic hinges
to cause structural failure [15- 20, 22], are selected by using a
branch-and-bound technique [17-20, 22]. Then, probabilities
of occurrence of the plastic collapsing are estimated, based on
the selected failure paths. Finally, a numerical example of a
jacket-type offshore platform is provided to demonstrate the
validity of the proposed method.
Modeling of Wave Loading
Probabilistic properties of wave loads acting on a jacket-
type structure composed of slender cylindrical members are
modeled
for
reliability assessment.
First,
a
force
dLk(t)
induced on
an
incremen tal element
dl
of a member
k
d ue
to
motion
of
water particles
is
estimated
by Morison's formula [23]
where
dL
k
(t)
p
dL
k
{t) = (
[
/iC
D
pD v
p
\v
p
\
+ C
M
pAv„)
dl
— force acting in the direction norma l to the member
= water density
= instantaneou s velocity of the water particle, normal
to the longitudinal axis of the member
= corresponding acceleration of the water particle
= diameter of
th e
member
= cross-sectional area of the memb er (= 7r/4 • D
1
)
27 0 / V o l . 1 09 , A U G U S T
1987
Transact ions
of
the ASME
j
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C
D
= drag coefficient
CM
= mass coefficient
Consequently, the total force L
k
{t ) on the member k is given
by
•A)
£.*( / )= a f l*(0
(2)
where l
k
= length of the member.
The center of action 4
c a
of the wave force is calculated by
- f '
I dL
k
(t)\/L
k
{t)
(3)
The problem in applying Morison's formula is to choose
the values of the coefficients
C
D
and
CM
for the situa tion
under consideration. These coefficients are dependent on
many factors , such as Reynolds num ber , Keulega n-Carpenter
number, the relative roughness of the member surfaces, etc.
[7], which are difficult to exactly estimate in practice. There
fore, C
D
and
CM
are treated as random variables in modeling
a wave load.
The water particle kinematics is estimated by using Stokes
third-order theory [24] , which formulates the velocity and
acceleration of the water particle as a function of the wave
height
H,
the wave period
T
w
,
and the water depth
d.
For a typical drag-dominated structure, the variation in the
wave force is much more sensitive to a wave height than to a
wave period. Consequently, the relationship between the wave
period T„ , and the wave height H is assumed here to be
deterministic, as given in the following:
T„
xH
(4)
where
a,
/3 = empirical consta nts.
On the other hand, the wave height of an individual wave
varies randomly and is assumed to be distributed in Rayleigh
form
Ah
P W = jp.
ex
P
2/r;
H,
2
(5)
where
Pii(h)
= probability density function
H
s
= significant wave height
What is important in connection with an extreme wave
loading problem is not the probability distribution of the
individual wave height / / , but that of the maximum wave
height //
m a x
during the postulated extreme sea-state. When
the individual wave heights are assumed to be independent
random variables, the probability distribution function
^Hmax(/0 of the maximum wave height, in a sea-state with a
significant wave height
H
s
and a duration of the sea-state
T,
is given by
PnJLh) = [P„(h)Y
(6)
where
PHQI) =
Jo
PH{II) dh
= probability distribution function of
an individual wave height
n
= num ber of waves in the duration of the sea-state
T,
i.e., n= [T/T
0
]
[•] = G auss 's nota t ion
T
0
=
mean value of the wave periods
I
hen, the expected value of the m axim um wave height 7/
max
is calculated by
E[H
m
^\
-r
• n • \P
H
{h)} -
l
p
H
{h) dh
(7)
The variance of H
m
C H n
is given by
= £ [ / / L x ] - \E[H„
J ]
2
(8)
Le t the maximum wave force L
k
(t) on the kth member be
a function
g
k
of the random variables 7/
m ax
,
C
D
and
C
M
L
k
(t) = g
k
(Y, t)
(9)
w he re Y = (7 , ,
Y
2
, Y,)
T
=
(7 /
m a x
,
C
D
,
C
M
)
T
.
Experimental results indicate that the drag and mass
coef
ficients are negatively correlated [25]. Consequently, it is
assumed here that the
C
D
and
C
M
follow a joint G aussian
distribution w hose probability density function is denote d by
pC
D
C
M
(yiyi)-
On the o ther hand , the max imu m wave he ight
is indep ende nt of the M orison 's coefficients. T hen, the me an
and variance of the wave force are calculated by
M i
t
</)
- / / /
al
k
u)
- H I
g
k
(Y, t)p
Ha m
(yi)Pc
D
c„(y2, yj) dy, dy
2
dy j (10)
g
k
(\, tfp
Hn
,Jyi)Pc
D
c
M
(y2, y
3
) dy, dy
2
dy
3
-W
t
(t)\
2
( ID
In general , i t is not easy to evaluate equations (10) and (11),
and thus a f i rs t -order-second-moment (FOSM) method is
applied to approximate them. The resulting expressions are
given as follows:
<
HL
k
= gk(flH
m:a
, MC
B
, M Q , )
3 f ^ „ . |
2
3 3
,•-. lay,
r--
+
m
*i
jdgk
'
\dYj
» *
YjPij
(12)
[13)
where
Y = the mean value vector of a ran dom variable vector Y
cry. = the standard dev iation of a ran dom variable 7,
p,j
= the correlation coefficient between the random vari
ables Yi a nd Yj
and the terms in the parentheses represent the partial deriva
tives of the function evaluated at their mean values.
Due to the complex nature of the function g
k
(Y, t), its
derivatives have to be determined numerically.
A utom at ic Ge ne r a t ion o f S tr uc tur a l Fa i lur e M od e
Consider a frame structure whose elements are uniform
and homogeneous and to which only concentra ted loads and
moments are applied. In such a frame structure, crit ical
sections where plastic hinges may form are the joints of the
elements and the places at which the concentrated loads are
applied. The following description is concerned with the case
when plastic hinges occur under combined load effects sub
jected to a bending moment and an axial force. The behavior
of mem bers are defined as shown in Fig. 1. Structural analysis
is performed by combining a plastic hinge method and a
matr ix method based on the d isplacement method [21].
Derivation of Reduced Stiffness Matrix and Equivalent
Nodal Forces. L et X, = (F
xi
, F
yh
M
zi
, F
xj
, F
yj
, M
zj
)
T
and
& :
=
(Vxi, Vyi, d
zi
, v
xj
, v
yJ
, 6
ZJ
)
T
denote the nodal force and
displacement vectors of the unit element
i,j,
e.g., the elemen t
number / in the local coordinate system shown in Fig. 2.
In order to avoid the difficulties of yield condition, the yield
Journal of Offshore Mechanics and Arctic Engineering AUGU ST 1987 , Vo l. 1 09 /27 1
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Bending
moment only
Fig.
1 Lineariz ed plasticity condition
F . , v .
XI XI
F . , v •
y j y j
Fig.
2 Nodal forces and nodal displacements
surface is assumed by a linearized function as shown in Fig.
1. Then, plasticity condition of
a
cross section is given in the
following form:
F
k
=
R
k
C
k
X, 0 (k=i,j)
(14)
In equation (14),
R
k
is the reference strength of the element
end k, which is taken to be a fully plastic moment, i.e.,
R
k
= OykAZpk
(AZ
pk
is the plastic section modulus of the
element end
k, a
yk
is the yield stress).
C
k
T
is a factor deter
mined by the dimension of the element k. Particularly, the
expression taking account of the interaction between the
bending moment and the axial force upon the plasticity
condition is given as follows:
C ,
r
= (A Z
pi
/A
pi
sign(F
xi
), 0, sign(M
z/
), 0, 0, 0) (14a)
C,
T
= (0, 0, 0,
A Z
PJ
/A
PJ
sign(F
xj
), 0, sign(M
z
,)) (14b)
where
A
pk
(k = / j) = cross-sectional area of the element end k
sign(-) = signof(-)
In the well-known plasticity condition of a plane-frame struc
ture subjected solely to bending moments is obtained by
putting the first term of C,
r
and the fourth term of C / equal
to zero.
The yielding condition of equation (14) is graphically illus
trated in Fig. 1. A solid line shows the failure criterion
considering the com bined load effect subjected to the bending
moment and the axial force, and a broken line shows the
criterion considering only the bending moment.
Next, the behavior of yielded portion follows the plastic
deformation theory because the perfectly elastic-plastic rela
tionship has been em ployed into the plasticity cond ition. The
relation between the nodal force vector X, and the displace
ment vector 6, of an elem ent including plastic hinges is derived
by using plastic deforma tion theory as follows [21]:
X, = k ,
(
">S, + X,<
(15)
where
k,
(p)
= reduced element stiffness matrix
X/
p>
= equivalent nodal force vector
The explicit forms of
k,
ip
\
and X,
<p)
are expressed
follows:
as
1 In case of an elastic element:
k,
fp)
= k,
(k, = elastic element stiffness matrix) X,
{p)
= 0 (16a)
2 In case of failure at the left-hand end:
k,<"'(=k,
i
) = k, - t C C / M C / l c G )
X/"»(=X,
i
) = ^ C A Q T c C )
3 In case of failure at the right-hand end:
k,
(
*>(= V ) = k, - k , q c / k , / ( C / k , q )
XV">(=X/) = / j ,k ,c, / (C/k,C,)
4 In case of failure at both ends:
k,^(=k,
LK
) =
k, -
[H ]
T
[G~'][H]
(16/))
(16c)
X,°»(=X,
L
*
[G -
1
] =
[//] =
[H ]
T
[G~
C/k,C, C/k,Q_
C,
r
k,
C/k ,
(\6d)
Automatic Generation of Safety Margins and Structural
Failure Criterion. Consider a frame structure with n ele
ments a nd at m ost 3 / loads applied to its / nodes. The failure
criterion of the (th element end is given by
Z,
= R, - C ,
r
X, S 0
(17)
Structural failure of a frame structure is defined as occur
rence of large nodal displacement due to plastic collapsing. A
criterion for structural failure is given in the following man
ner. When any on e element end yields, the internal forces are
redistributed to the element ends in survival and an element
end next to yield is determined. Repeating the processes,
when the element ends r
x
, r
2
, • •., r
p
-\ have failed, stress
analysis is performed once again and the element stiffness
equation is obtained as
X, = k,
{p)
& , + X,
(
"> (18)
The reduced element stiffness matrixes are evaluated for all
the failed elemen ts, and they are assembled to have the total
structure stiffness matrix
K
(
"'d = L + R
(
">
(19)
where
d: total nod al displacem ent vector referred to the global
coordinate system
K
<p)
= £ =i T /k /^ 'T , : reduced total structure stiffness ma
trix
2 7 2 / V o l . 109, AUG UST 1987 Transact ions of the ASME
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T,: transformation matrix
L: vector of the external loads
R"" = - X"=>
Ti
T
Xi
ip)
:
equivalent nodal force vector re
ferred to the global coordinate sys
tem
Finally, the nodal force vector X, of the
tth
member is given
by
X, = b,
(
">(L + R<">) + X,«" (20)
where b,""
=
V 'T . r j K , " " ] - '
Now that the element ends
r
x
, r
2
,
. . . , and
r
p
-i
have failed,
the safety margin
of
the surviving element
end
i (element
number
/) is
obtained
by
substituting equation
(20)
into
equation (17)
Z,
(
">
=
R,
+ C W £
T^X*'")
-
X,
(
"»)
k-l
- CV ' L
1
ml
Ri + 2J
a
ir
k
Rr
k
—
L
bijLj
k-
I j=1
(22)
where
a
irk
a nd
b
u
are the coefficients resulted from resolution
of the vectors into their components.
Occurrence
of
the plastic collapse
is
determined by inves
tigating
the
property
of
the to tal struc ture stiffness matrix
[K
ip,,)
] and the total nodal displacement vector d. For exam
ple, when the element ends up to some specified number p
q
,
e.g., element ends u, r
2
,
. . . , and
r
Pq
, have failed and either
the total reduced structure stiffness matrix [K
(p
«
)
] or the total
nodal displacement vector
d
satisfies the following condition,
structural failure results:
|[K<*>]|/|[K<°>]| Sc ,
| | H < 0 > | | / | | J < / > , ) || < «„
(23a)
(23b)
where superscripts (p
q
) and (0) are used to denote the p
q
X\\
failure stage and the elastic condition, respectively. No rm
|| || is norm of the total nodal displacement vector d. t
x
and
c.2
are specified constants for determining the plastic collapse.
By using the foregoing equation,
a
criterion
of
structural
failure is given by
Z<;» ^ 0 (p=l,2,...,p
q
) (24)
If there are any failed elem ent ends
r
p
,
which have their
coefficients
a
rp
,
p
equal
to
zero
in the
safety margin Z[
PQ )
of
the last yielded element end r
Pq
, i.e.,
a
Wp
= 0 (25)
they are the redundant element ends which
do not
directly
contribute to o ccurrence of the plastic collapse. Alternatively,
those element ends are called essential w ithout which no
plastic collapses are formed. Further,
it
should be noted that
the plastic collapse
of
any
one
element
end of a
statically
indeterminate frame structure does
not
necessarily result
in
the total structural collapse. A minimum set of plastic hinges
is defined as a set of the plastic hinges which constitutes a
failure path including no redundant plastic hinges [22],
Automatic Selection of Proba bilistically Dominant
Failure Paths
There are too many failure paths in a highly redundant
structure [22]
to
generate
all of
them, which necessitates
a
procedure for selecting only the probab ilistically significant
failure path s
[ 15-20,22].
Efficient methods by using
a
branch-
and-bound technique have been proposed [17-19,
22] and
this paper adopts the procedure given in the following.
Branching Operations. These operations are to select the
plastic hinges such that stochastically dominant failure paths
may be obtained. An element end (called here section for
simplicity) is selected as a plastic hinge at the pth failure stage
based on the criterion tha t the joint probability to fail is to be
the largest, i.e., the section r
p
to be selected at the pth stage is
given by
nzz, < o]
max
P[Z)
l>
2 0]
for
p
=
1
(26)
P[(z
(
r
\\
q)
£ 0) n (z\ < 0)]
= max
P[(Z
(
r
\\
q)
< 0) n (Z*;' g 0)]
for p
S 2 (27)
(21) where
I
p
= the set of sections i
p
to be selected at the pth failure
stage
Zj ' '
=
safety margin
of
section
/, at the
first failure stage,
i.e., when no plastic hinges exist in the struc ture
Z\
p)
= safety margin
of
section i„
at
the
pth
failure stage,
i.e., after formation
of
plastic hinges
at
the sections
r
u
r
2
, ...
,
and r
p
_, (p
g
2)
The lower and upper bounds,
P\ ^
q){L)
and
P\^
qW
)
°f
tn e
probability
P\£
q)
of a partial failure path up to the
pth
(p
§ 2) failure stage is evaluated as [17-19, 22]
p p < p p) p
r
fp(l)(L) =
r
fp(q)
r
o</>>
n (Z<;>
9)
g 0)
= Pfpiqxu) (28)
'««-.r/
[Z
*'-
0)n(Z
<
j e [2 , . . . , p |
0)
(29)
Ptlw =
maxjo ,
P[Z
q)
g 0]
r,(9)
7
(1)
- P[(z^
q)
̂0) n z
r
f
U)
> 0)]
I
min^'jU)
J = 3
/>[(z«;»„ s o j n (z > o)])| (30)
where subscript p denotes the failure stage and subscript q
is
introduced to indicate the particular selected failure path. By
repeating the selecting process, a sequence of plastic hinged
sections to form
a
plastic collapse, e.g., r,, r
2
, ..., and r
Pq
,
is
found.
The maximum P
(m
of the lower bounds of the selected
complete failure path prob ability is calculated [22]
P
fP
M =
max
' fpML)
(31)
Consequently, P
/pM
is updated when a new complete failure
path is found and its failure probability is larger than the
previous P/
pM
- The branching operations are terminated when
no sections are left for selection.
Bounding Operations.
These operations are
to
select
the
sections to the discarded. These are the sections deleted at the
pth failure stage
P[ Z
{
g
0]
< 10"
r
for p=\
(32)
IfpM
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of
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P[(z
q)
s 0) n (Z<;> s 0)]
r pM
< l(T
r
for p^2 (33)
From th is ,
it is
conc luded tha t
the
neglected failure paths
are those which have the failure probabilit ies smaller than
\0-*-P
JpM
[22}.
The probability of occurrence
P/
for the failure mo de
corresponding to a selected failure path is estimated by using
the safety margin
Z\''
q)
of
the last plastic hinge. That
is
Pr = pizy
s
oi
(34)
The expressions to est imate the collapse probability of the
total structure are given in the Appendix .
N u m e r i c a l E x a m p l e s
A jacket-type offshore platform shown
in Fig. 3 is
chosen
for numerical examples. The d imensions are given in Table
1.
Est imat ion
of
wave loadings
is
first given
and
then proba
bilistic collapse analysis
is
carried
out.
Wave Loadings. The extrem e wave forces induced on the
me mbe rs
are
calculated
for a
specified sea-state with
a
signifi
cant wave height
H
s
and a
dura t ion
of the
sea-state
T. Nu
merical data conc erned are listed
in
Table 2. First , com pariso n
of the mean
and
variance
of
the wave force
is
made between
the integral method, equations (10) and (11), and the F O S M ,
equat ions (12) and (13). The calculated values for the m e m
bers
1, 3, 5 and are
given
in
Table
3. It is
seen that both
methods yield almost the same results. Consequently, the
FO SM
is
used hereafter
for
evaluating
the
probabilistic char
acteristics
of
the wave forces.
Next , the effect of the phase between the structure and the
20 m
Fig. 3 Jacket structure
Table 1 Num erical data of jacket structure
Member
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Element end
number
1 ,
(2), 3, 4
5 ,
(6 ), 7, 8
9 , ( 1 0 ) , 1 1 , 1 2
1 3 , ( 1 4 ) , 1 5 , 1 6
1 7 , ( 1 8 ) , 1 9 , 2 0
2 1 , ( 2 2 ) , 2 3 , 2 4
2 5 , ( 2 6 ) , 2 7 , 2 8
2 9 , ( 3 0 ) , 3 1 , 3 2
3 3 , ( 3 4 ) , 3 5 , 3 6
3 7 , ( 3 8 ) , 3 9 , 4 0
4 1 , ( 4 2 ) , 4 3 , 4 4
4 5 , ( 4 6 ) , 4 7 , 4 8
4 9 , ( 5 0 ) , 5 1 , 5 2
5 3 , ( 5 4 ) , 5 5 , 5 6
5 7 , ( 5 8 ) , 5 9 , 6 0
Outside
diameter
D.
m
i
0 .76
0 .70
0 .36
0 .46
0 .36
0 .46
0 .36
0 .46
Cross sectional
area
A
.
m
Pi -
0 .0810
0 .0638
0 .0154
0 .0200
0 .0154
0 .0200
0 .0167
0.0247
Moment of
inertia
I. m
5 . 5 8 7 x l 0 "
3
3 . 7 4 6 x l 0 "
3
2 . 3 9 6 x l 0 ~
4
5 . 1 4 7 x l 0 "
4
2 . 3 9 6 x l 0
- 4
5 . 1 4 7 x l 0 "
4
2 . 5 9 7 x l 0 ~
4
6 . 2 9 6 x l 0 ~
4
Mean value of
reference strength
R. kNm
i
5286 .0
3842 .0
476 .9
798 .5
476 .9
798 .5
517 .8
980 .4
Young's modulus E = 210 GPa
Mean value of yield stress a . = 276 MPa , Coeff. of variation CV
n
Iv
J
Yi
0.08
Mean value of foundation pile (61,62) capacity R„ = 13070 kN Coeff. of variation CV = 0.08
Correlation coefficients : p.. = 1 for the element ends in the same type of members
I'd
while
p.
= 0
for the
different types
of
members.
I'd
Number in bracket designates an intermediate element end which does not fail.
274
Vol. 109 , AUGU ST
1987
Transac t ions
of the
ASME
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wave is examined on the resulting wave forces. Table 4 shows
the typical results which are obtained for the various positions
of x
e
of the wave crest relative to the central line of the
platform. It is concluded that, although the phase which
produces the maximum wave force differs from one member
to another, the sum of the mean values of the horizontal
forces acting on all the me mbers is maximum when the wave
crest is approach ing the place of approximately A/40 (X is the
wavelength) in front of the central line of the structure .
Probabilistic Collapse Analysis.
Probabilistic collapse
analysis is carried out for the offshore platform of Fig. 3,
Table 2 Postulated extreme sea-state and Morison's coeffi
cients
Significant wave height
duration of sea-state
Wave period
C
D
C
M
Mean value
Coefficient of
variation
Mean value
Coefficient of
variation
Correlation between
C
D
an d
C^
H
s
T
T
w
^D
CV
C
^M
CV
C
P
C C
D
L
M
13 (m)
1 ( h o u r )
T = 4 . 51 - H
0 - 5 5 9
w
(sec) (m)
0 . 7 5
0 . 3
1 .8
0 . 3
- 0 . 9
which is exposed to the postulated extreme sea-state given in
Table 2. The Morison's coefficients are also specified in the
table. The resulting wave loads are calculated as described in
the previous section and listed in T able 5. The statistical data
of the deck loads L
7
, L
8
are added in the table. The strengths
of the members are given in Table 1. The capacities of the
foundation piles 61, 62 are modeled as the elements which
are fixed at one end with their rigidities very large and fail
when the applied axial forces reach the specified values, as
given in the table. All the random variables are assumed to
be distributed normally with the parameters given in Tables
1 and 5.
Typical failure paths selected by the procedure in the pre-
Table 3 Comparison of
the
methods for calculating the mean
and variance of the wave loads
\
Load
\ .
L
1
h
Ratio of 'pro
cessing time
Integral method
_ * **
L. CV
Tl
K
Lk
3 5 4 . 4 0 . 4 2
1 6 7 . 4 0 . 4 2
1 2 7 . 1 0 . 4 3
5 0 0
FOSM
k Lk
3 6 8 . 1 0 . 4 2
1 8 1 . 5 0 . 4 1
1 4 2 . 7 0 . 4 1
1
* : Mean value of load ( kN )
** : Coefficient of variation of load
x A = -1/40
Table 4 Effect of the position of the wave crest relative to the central line of the platform
N .
Load
\ ^
Position of the wave crest ( x /X )
- 1 / 1 0 - 1 / 2 0 - 1 / 4 0 0 1 / 4 0 1 / 2 0 1 / 1 0
L,
1
L
2
i .
3
L
4
L
s
£ .
6
L
1 2
L
n
1 3
2 6 2 . 6
#
( 0 . 3 3 )
$
1 3 2 . 8
( 0 . 2 7 )
1 4 9 . 5
( 0 . 3 3 )
8 8 . 2
( 0 . 2 3 )
1 3 2 . 6
( 0 . 3 2 )
7 8 . 4
( 0 . 2 1 )
3 5 . 0
( 0 . 2 5 )
2 5 6 . 9
( 0 . 4 0 )
3 6 0 . 5
( 0 . 4 0 )
2 7 9 . 2
( 0 . 3 5 )
1 8 3 . 5
( 0 . 3 8 )
1 4 6 . 3
( 0 . 3 2 )
1 4 9 . 0
( 0 . 3 3 )
1 2 0 . 7
( 0 . 3 0 )
9 2 . 1
( 0 . 2 6 )
2 4 5 . 8
( 0 . 4 4 )
3 6 8 . 1
( 0 . 4 2 )
3 3 4 . 4
( 0 . 3 8 )
1 8 1 . 5
( 0 . 4 1 )
1 6 9 . 6
( 0 . 3 6 )
1 4 2 . 7
( 0 . 4 1 )
1 3 7 . 6
( 0 . 3 3 )
1 3 9 . 0
( 0 . 3 3 )
2 0 5 . 0
( 0 . 4 6 )
3 4 1 . 2
( 0 . 4 4 )
3 6 5 . 7
( 0 . 4 0 )
1 6 5 . 3
( 0 . 4 3 )
1 8 2 . 8
( 0 . 3 8 )
1 2 6 . 6
( 0 . 4 4 )
1 4 7 . 5
( 0 . 3 6 )
1 8 9 . 4
( 0 . 3 7 )
1 4 8 . 9
( 0 . 4 9 )
2 8 4 . 1
( 0 . 4 6 )
3 6 4 . 7
( 0 . 4 2 )
1 3 6 . 9
( 0 . 4 6 )
1 8 2 . 6
( 0 . 4 0 )
1 0 2 . 4
( 0 . 4 8 )
1 4 8 . 1
( 0 . 3 9 )
2 3 2 . 4
( 0 . 3 9 )
8 8 . 3
( 0 . 5 6 )
2 0 7 . 5
( 0 . 4 9 )
3 2 9 . 4
( 0 . 4 4 )
1 0 0 . 5
( 0 . 5 1 )
1 6 8 . 1
( 0 . 4 3 )
7 3 . 1
( 0 . 5 6 )
1 3 8 . 4
( 0 . 4 2 )
2 5 8 . 4
( 0 . 4 1 )
3 4 . 7
( 0 . 9 1 )
4 6 . 1
( 0 . 9 0 )
1 8 5 . 7
( 0 . 5 1 )
2 5 . 8
( 1 . 0 5 )
1 0 5 . 1
( 0 . 5 1 )
1 4 . 0
( 1 . 6 9 )
9 3 . 0
( 0 . 5 0 )
2 4 0 . 1
( 0 . 4 3 )
- 1 9 . 7
(-0.57)
Mean value of load L-.
(kN)
$ -. Coefficient of variation of load ( CV )
Journal of Offshore Mechanics and Arctic Engineering AUGUST 1987, Vol. 109 / 275
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Table 5 The wave loads induced by postulated sea-state and
Morison's coefficients, and deck loads
Kind of loads
Wave load
Deck load
No.
L
y
h
h
h
h
h
h
L
W
hi
V
V
L
u
hs
he
hi
h
h
Mean value Coeff. of variation
L . kN
3
368 .1
334 .1
181 .5
169 .6
142 .7
137 .6
4 . 5
5 .5
3 .7
139 .0
205 .0
7 2 . 3
8 8 . 9
5 6 . 9
59 .9
2490 .0
2490 .0
CV
T
.
h
0 .42
0 . 3 8
0 .41
0 .36
0 .41
0 . 3 3
1.74
0 .68
0 .45
0 . 3 3
0 .46
0 .31
0 .45
0 . 3 4
0 .40
0 .10
0 .10
Correlation
coeff.
P T ^
• = 1.0 {i , jell,.. . , 6 , 1 2 , . . . ,1 7} )
= 1.0 (i,jell,8})
= 1 .0 ( i
i t
7 ' e { 9 , 1 0 , l l } )
= 0 . 0 (i,jel o t h e r s })
Table 6 Collapse modes and failure probabilities
E, - 0 .001, E , - 0.05 , T - 4.0
Failure paths Failure probabilii
( 3 5 , 3 6 , 3 3 )
( 3 5 , 3 6 , 2 5 , 3 3 )
(35 ,36 ,31 ,19 ,33)
(35,36,31,33)
( 3 5 , 3 6 , 3 1 , 1 9 , 2 9 , 3 3 )
( 3 5 , 3 6 , 3 1 , 2 5 , 3 3 )
( o t h e r s )
0.8701-10
0.8498*10'
0.8149x10'
0.7846x10'
0.7728«10'
0.7723x10'
<0.75x10'
(35,36,31,25,29,32,33) 0.6684x10'
(35,36,31,19,29,28,32,27,33) 0.6285x10'
(35,36,31,19,29,28,32,33) 0.6179x10'
(35,36,31,25,19,29,32,33) 0.6059x10"
(35,36,31,19,29,28,32,27,23,33) 0.5638x10'
(35 ,36 ,31 ,25 ,19 ,17)
(35 ,36 ,31 ,25 ,29 ,19 ,17)
0.1461*10"'* [ 3]
0 .1306x l0 "
4
[ 5]
[ 1]
[ 1]
[ 1]
[ 1]
[ 1]
[ 5]
[10 ]
[ 2]
[ 2]
[ 1]
[ 7]
[ 3]
D- l . ( 35 ,36 ,31 ,25 ,19 ,29 ,32 ,17)
0.1185x10
4
[ 7]
E- l. (62)
(penetration of foundatii
pile)
0.7125x10 ' [ 1]
mw, wj7'
Computation time (sec)
The figures in the brackets indicate the numbers of selected failure paths .
vious section are given in Table 6. In the table, the failure
paths which have the same minimum sets of essential plastic
hinges are integrated in the same group and the listed paths
correspond to those which have the maximum failure proba
bilities. Note that the failure probabilities are calculated with
the safety margins of the last plastic hinges, i.e., equation (34),
and given in the second column.
It is seen that the dominant failure modes of this jacket
structure are those plastic collapses which are formed, trig,
gered by failure of the brace members in the top story. The
failure probabilities are relatively small for the failure modes
which include failure in the c olumn mem bers. The probability
of failure du e to the penetration of the foundation pile is very
small in this example.
Conclusion
The methods are proposed for the probabilistic collapse
analysis of the offshore structure. At first, the wave loads
induced by the extreme sea-state are estimated by using Stokes
third-order theory and Morison's formula. Second, a method
is presented for the evaluation of plastic collapsing taking
account of the combined load effect of the bending moment
and the axial force, and for the generation of the safety
margins. Third, probabilistically dominant collapse modes
are systematically selected by using a branch-and-bound
method. Finally, the proposed methods are successfully ap
plied to the collapse analysis of a three story jacket-type
offshore platform. For the example structure, it is concluded
that the collapse modes triggered by failure of the brace
members in the top story are probabilistically dominant and
that the other modes, such as the failure mo des related to the
column members and the foundation piles, have the proba
bilities of failure which are relatively small.
Acknowledgment
The authors would like to express their thanks to Prof. K.
Taguchi for his encouragement and to Mr. S. Katsura for his
help in the numerical calculations. A part of this work is
financially supported by a G rant-in -Aid for the Scientific
Research, the Ministry of Education, Science and Culture of
Japan. All the computations are processed by using ACOS
700 at the Computer Center of the University of Osaka
Prefecture.
References
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3 Bouma, A. L., Monnier, T., and Vrouwenvelder, A.,"Probabilistic Re
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Aug. 28-31, 1979, pp. 521
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4 Bea, R. G ., "Reliability Consideration in Offshore Platform Criteria,"
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he
ASCE,
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he Structural
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Sept. 1980, pp. 1835-1853.
5 Anderson, W. D., Silbert, M. N„ and Lloyd, J. R., "Reliability Procedure
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6 Thoft-Christensen, P., and Baker, M. J.,
Structural Reliability Theory
an d its Applications, Springer-Verlag, 1982, pp. 203-207.
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9 Schueller, G . I., "Re liability Based Optim um Design Offshore Plat
forms," International Journal on Probability and Statistics in Engineering
Research an d Development,
Marcel Dekker, International, New York, Vol. '•
1983.
10 Mart indale, S. G ., and Wirsching, P. H., "Reliability-Based Progressive
276 / Vol. 109, AUG UST 1987 Tran sac t ions of the ASME
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1982, pp. 525-540.
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A P P E N D I X
Collapse Pro bability of the Total S tructure
When a sequential failure of the element ends r
u
r
2
,
. . . , and r
p
turns the structure into a plastic collapse,
the probability of occurrence of the complete failure
path is exactly calculated by
pip,,)
fp(q)
ri (z?> si 0)
/=
(35)
Consequently, the collapse probability Pf of the total
structure is given by
u n (z „ ^ 0))
q \l= 1
(36)
where the union is carried over all the failure paths.
When the proposed selection procedure is applied,
equation (36) is evaluated as [22]
< j e * A i = l
=
r
f
u
q£X
c
\l=i J qEX, \l-\
g P
u
+ E
(37)
where the union with respect to q means to be taken
over all the selected complete failure paths X
c
or all the
discarded failure paths X, an d E is the contribution of
the discarded failure paths.
Journa l of O f fshore Mechan i cs and Arc t i c E ng in eer i ng AUGU ST 1987 , Vo l . 109 /27 7
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