probabilistic collapse analysis of offshore structure

8/18/2019 Probabilistic Collapse Analysis of Offshore Structure

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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

Copyright © 1987 by ASME wnloaded From: http://offshoremechanics.asmedigitalcollection.asme.org/ on 04/11/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use



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




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

Journal

of

Offshore Mechanics and Arctic Engineering AUGU ST 1987 , Vo l . 109 /27 3




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




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




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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Reliability Function Model," OTC Conference Paper 3028, 1977.

2 Moses, F., "Safety and Reliability of Offshore St ructure s," eds., I.

Holand, et al. Safely of Structures under Dynamic Loading, Vol. 1, Norwegian

Institute of Technology, Trondheim, 1978, pp. 546-566.

3 Bouma, A. L., Monnier, T., and Vrouwenvelder, A.,"Probabilistic Re

liability Analysis,"

BOSS 79,

Aug. 28-31, 1979, pp. 521

-542.

4 Bea, R. G ., "Reliability Consideration in Offshore Platform Criteria,"

Proceedings

of

he

ASCE,

Journal

of

he Structural

Division, Vol. 106,

No. ST9,

Sept. 1980, pp. 1835-1853.

5 Anderson, W. D., Silbert, M. N„ and Lloyd, J. R., "Reliability Procedure

for Fixed Offshore Platforms,"

Proceedings of the ASCE, Journal o f the Struc

tural Division, Vol. 108, No. ST11, Nov. 1982, pp. 2517-2538.

6 Thoft-Christensen, P., and Baker, M. J.,

Structural Reliability Theory

an d its Applications, Springer-Verlag, 1982, pp. 203-207.

7 Soares, C. G ., and Moan, T., "O n the Uncertainties Related to the

Extreme H ydrodynamic Loading of a C ylindrical Pile," ed., P. Thoft-Christen

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Martinus Nijhof, 1982, pp. 351-364.

8 Karadeniz, H., Vrouwenvelder, A., and Bouma, A. C , "Stochastic

Fatigue Reliability Analysis of Jacket Type Offshore Structures," ed., P. Thoft-

Christensen, Reliability Theory and its Application in Structural and Soil

Mechanics, Martinus Nijhof, 1982, pp. 425-443.

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




Fatigue Collapse," Journal of Structural Engineering, ASCE, Vol. 109, No . 8,

Aug. 1983 , pp. 17 92-18 11 .

11 Croha s, H. , Tai , A. , Hach emi, V. , and Barunoi n, B. , "Reliability of

Offshore S truc tures under Extreme En vironmenta l Loading, " OTC Conference

paper 4826, 1984.

12 Furnes, O. , and Sele, A. , "Integrity of Offshore Structures ," eds. , D.

Faulkner, M. J. Cowling, and P. A. Friege, Implemen tation of Offshore Struc

tural Reliability, Applied Science Publication s, London an d New Jersey, 1981.

13 Chrys santho poulo s, M., and Skjerven, E. , "Progressive Collapse Analysis

of a Jack-up P la t form in Intac t and Damaged Co ndi t ion , " DnV, Tech. R epor t ,

No. 83-1026, 1983.

14 Ueda , Y„ Rashed, S. M. H. , and Nakach o, K. , "New Efficient an d

Accurate Method of Nonlinear Analysis of Offshore Tubular Frames,"

Pro

ceedings of the 3rd International Symposium on Offshore Mechanics an d Arctic

Engineering, ASME, Vol . 1 , 1984, pp. 260-268.

15 Moses, F. , "System Reliability Deve lopm ents in Structural Engin eering,"

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Probabilities of Ductile Structures by the /3-unzipping Method," Institute of

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Nijhof,

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

probabilistic collapse analysis of offshore structure

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