Linearized mechanical model to evaluate reliability of offshore structures

Structural safety ELSEVIER

Structural Safet~ 17 (1995) 167-193

Linearized mechanical model to evaluate reliability of offshore structures A.M. M o h a m e d * , M. L e m a i r e Lx~RAMA, Frem'll Institute o/'Advanced Mechanics, F-63175 Aubibre Cedex, France

Abstract In this paper, an efficient reliability method is proposed for space framed structures. It is based on linearization of the mechanical behaviour. The elastic-plastic behaviour of material is piecewise linearized by a plastic criterion taking into account the interaction of internal forces. The buckling effect is represented by using an equivalent negative strain-hardening, introduced to the material model. In order to identify the negative strainhardening parameters, a powerful method is given. The proposed model also takes into account the unloading of members and the behaviour at criterion singular points. The structural reliability is estimated by the safety margin concepts. Numerical examples show the high performance of the model, verified by a good precision and a very low computational time. Kevwords. System reliability: Nonlinear systems: Plasticity: Buckling; Post-collapse behaviour; Offshore structures

1. Introduction Offshore jackets have important reserve strength due to a high degree of redundancy. Platforms can resist design loads even in a deteriorated state. Reliability analysis of such structures must take into account the inelastic behaviour effects. In the post-elastic range, the modelling of structural behaviour is of great importance because it defines the redistribution of internal forces, and consequently, the failure modes. The accuracy of reliability estimation depends on the way in which the real behaviour is represented. Knowing that this behaviour is generally nonlinear, the mechanical model and the reliability model become interdependent. An accurate modelling of real behaviour implies the use of nonlinear incremental methods. In this case, the reliability can be calculated by sampling techniques which are inconvenient for practical use. An alternative is given by the linearization of mechanical behaviour allowing the explicit description of safety margins [ 1,2]. However, the linearization process leads to significant loss of accuracy. To optimize the relation between computational time and accuracy, * Corresponding author. 0167-4730/95/:9.)9.50 @ 1995 Elsexier Science B.V. All rights reserved SSDI 01 67 -4730( 95 ) 00009-7

168

A.M. Moltam¢'d. M. Lematre/Strt,'tural Sq/bty 17 (1995) 167-193

we should seek a linearized method having the advantage of a high degree of accuracy, compared with complete nonlinear models. For large scale structures, the failure path approaches are extremely efficient. In these approaches, the structural failure is reached by suitable combinations of discrete component failures. Between two successive component failures, the mechanical behaviour is linearized. For trussed structures, the elementary way of linearization is given by the well-known "member replacement method" [3-5]. In these models, the failed members are eliminated and replaced by residual forces. This approach has been extended to the case of members having residual stiffness [6,7], and to the case of nonproportional loadings [81. In the case of framed structures, Thoft-Christensen and Murotsu [9] proposed a reliability method based on the generalized plastic criterion concepts. The failed element is replaced by a constant stiffness matrix and a constant vector of residual forces; this hypothesis was the base of many reliability softwares, such as ELFIAB 110] and ARPEJ [ I 1 ]. While the way of modelling truss members is very satisfactory, the accuracy of the beam-column models in framed structures is very low. The weakness of the member replacement method lies in its incapacity to represent adequately the post-critical behaviour of frame members. In order to meet realistic structural modelling, an improved method is suggested herewith, by the generalization of the concepts of Thoft-Christensen and Murotsu [9]. With respect to existing methods, the proposed method has the advantage of realistic plastic criterion with updated strainhardening, taking into account the effect of compression member buckling, cross-section unloading and singular point behaviour. This method represents a very' good compromise between realistic structural modelling and reasonable computational cost. The elastic-plastic behaviour is represented by the well-known plastic flow theory [12]. The interaction between axial force and bending moments is taken into account by a linearized plastic criterion with updated flow rule. During buckling, the behaviour is modelled and linearized by using an equivalent negative strain-hardening, representing the loss of member axial capacity. This negative strain-hardening allows us to incorporate the buckling phenomenon in the elastic-plastic model. The behaviour at section unloadings and at criterion singular points is taken into account. The joint failures can easily be incorporated in the model by using the joint plastic criterion instead of the cross-section criterion. An efficient method of identification of negative strain-hardening parameters is proposed by the analysis of member buckling curves. Hence, the mechanical model is piecewise linearized without significant loss of accuracy,, compared to complete nonlinear mechanical models. In structural systems, random variables are divided into external variables, as applied loads, and into internal variables, as component strengths. In case of linearized systems, the internal forces are linear functions of random variables. In order to separate the effect of the contribution of each random variable, an analytical integration of generalized plastic deformations is performed all over the loading history. The failure probability of each structural component is calculated by describing their safety margins in function of random variables and influence coefficients. The system reliability is obtained by combination in series and in parallels of component failure modes.

2. Structural nonlinear behaviour

Generally, the mechanical behaviour shows material and geometrical non-linearities. The analysis of such a behaviour can be obtained by classical nonlinear methods using many increments and

A.M. Mohamed. M. Lematre/Slructural &4fi'ty 17(1995) 167-193

169

iterations. In reliability context, safety margins are implicit and a complete nonlinear analysis is necessary to give the response for only one random sampling. Knowing that each nonlinear analysis is very costly, the treatment of the whole response spectrum becomes unadapted for practical use. Another way to perform mechanical analysis is given by the linearization of the structural behaviour. This reduces the computational cost and makes the mechanical response independent of random phenomena. But the inconvenience of this approach lies in the low precision with reference to "real" behaviour. In this study, we present a reliability method having a good compromise between accuracy and efficiency. The proposed method gives the mechanical response in a piecewise linearized form. It takes into account the interaction between internal forces as well as the effect of the buckling phenomenon, member unloading and singular point behaviour. And, it also allows the separation of influences of internal and external random variables (i.e. loadings and strengths). In order to apply the plastic flow theory for cross-sectional behaviour, we have to introduce the following assumptions : ( 1 ) The behaviour remains pure elastic until reaching the plastic criterion where it becomes completely plastic. (2) The generalized deformation increment can be divided into elastic and plastic parts. (3) The plastic criterion should be a convex function and the normality law must be respected. (4) The cross-section unloading from the plastic criterion is pure linearly elastic.

2.1. Inleractiov diagram ({/ tubular cross-sections [ 1.7/ Steel tubular cross-sections are widely used in offshore engineering. We give great attention to this kind of cross-sections; however, the method remains valid for other kinds of structures. For tubular space frames, the interaction between axial force and bending moments is considered. The effect of shear forces and torsional moment on the plastic behaviour is neglected. Let N be the axial force and, M, and ,4//_ the bending moments respectively' of the transversal axes y and z. The real plastic criterion, represented m Fig. 1, can be given by the following relationship:

f(N.M,,M:,a)=

I

Ms

Me, cos tg i

M.

,, sin (IMp)) tg Mp

. . . . .

- -

rr N [ COS

2 Nt,

--

a

where N~,, Mp are respectively the plastic cross-sectional capacities under pure axial force and bending moment; a is the strain-hardening magnitude; I I is the absolute value operator. In order to linearize the cross-sectional behaviour, a plastic criterion constituted by 24 plane surfaces is proposed. Figure 2 shows the criterion in the generalized stress space. It is identified by three parameters, y, ( and r/. The choice of these parameters allows us to represent different shapes of cross-sections. In the case of tubular members, suitable values of these parameters are found to be: y = 0.25, ( = 0.03 and r/ = 0.40. The general form of the criterion is given by the following relationship:

4.M. Mohanwd, M. LemairelStrucmral Safety 17 (1995) 167-193

170

A

+1

My my- Mp

/-

A

z

-/

'

/-

FI--

N Np

>. /

> Mz

Y

X

mz

-M ~

[Zig. I. Interaction diagram of tubular cross-sections. f=

J , C Sign ( M ) ~7_CSign( M: ) J,,,Sign(N) .M,+ -M.+ .N-(I M:,, M:,: N:,

+a

(1)

with

C=

1

y(f,,

1 - ( + T/J,:~

where Sign(.) is the sign o p e r a t o r and ,_7,, ,7-, J,,, are indexes defining the equaUon o f the c o n s i d e r e d plane surface (their values are "'0" or " 1 " ) . T h e s e values are given in Table 1, w h e r e n = N / N v, m>. = M , , / M > , m _ = M://Mm and D = ( 1 - I n ')//l 1 - Y ) . In the case of isotropic strain-hardening, the ranges o f generalized stresses are multiplied by the coefficient ( 1 + a ) .

2.2. Elastic-plastic fornlulatiotl In the case of space structures, each node has six d e g r e e s - o f - f r e e d o m and six c o r r e s p o n d i n g internal forces. C o n s i d e r i n g an e l e m e n t with ends i and ,j, the nodal actions are n o r m a l forces ( N , N / ) , shear forces ( k i , , F:, F,), F:, ), torsional m o m e n t s ( M , , , M , j ) and b e n d i n g m o m e n t s (M~., M=i, M : j , M : j ) , see Fig. 3. Thus, the vector o f nodal forces {/~;.} is defined by

{F,,}'=[N,

I[,, I"

/~I,, M,,. M:,, N,, F,/

F:,,

M,.i M,,j M:i ].

Tablc 1 Indexes for diffcrcnl plancs Plane Range of pl

Ranges of m, and m

1

y<]n

0 ~i m=

2

y <: ~t < 1

< I

Lf,

J,.

,.7:

I~ 711)

1

1

0

rid "~ m: ],4 t 1 ~)1) ~D
I

1

1

0

1

o

1

0

0

1

1

0

0

1

0 <] m, i.<- 77I)

!: r

4

o !l.

5

0 < Ji < ?

0 < m: !< rI <5~ m: I< (1 - 31) ~ "i ,,,, i<_ (1

6

0 ~ n i Y

I)
-

~1)

(2)

A.M. Mohamed, M. Lemaire/Structur, l Sc4fety 17 (1995) 167-193

1 Amy

+1A my

mz

q~ ql (1

171

My

=

-1

+1 ~1~

'+1"

In

-1

N " = Np

mz=lp

~ Z~

~.'~

"~_.__.h~" (1"~) Fig. 2. Linearized plastic criterion.

v,.i

" .... M

vy.i [

.

_. ~

Mx Fx Mzi

e

Fig. 3. Element nodal fbrces.

In the case of elastic behaviour, the element nodal forces {Fe} are related to the generalized nodal displacements {@} by the stiffness matrix [ K,. ]. The equilibrium relationship is as follows: {F,,} = [ K , I . {q~,}

(3)

with

{qe}t : [tll

l' I

w,

O~l

0~1

~:i

uj

l':

w~

O~j

Okj

] O:j]

where uk, z,~.,wk are respectively the displacements of node k (i.e. i or j ) according to directions x, y and z ; O,k, 0,~, O:k are respectively the rotations of node k around the x-, y- and z-axes. For yielded elements, the plastic criterion can be reached at one or both ends. At ends i and j, the linearized plastic criteria f , f j are written in the following simplified form: f , = {.,~'~}' - {F.} - N:,(I + a,),

(4a)

,4.M. Mohamed, M. Lemaire/Structural SqfeO' 17 (1995) 167-193

172

.[, = {.,%';}'. {F,.} -- Nr,(1 + a,)

(4b)

where the vectors {.&]}, {-'\'7~} are respectively perpendicular to the plastic criteria at i and j. They are given by

J,,,Sign( N,) 0 0 0 J.,.iC N~,Sign( M.,, )

0 0 0 0 0 0 ~,i Sign ( N j ) 0 0 0 (7~jCNpSign ( M,. ) Mp, JzjC N r,Sign ( M:e )

Mrh

(f.:,CN l, Sign ( M : ) {:'4}

=

l~4r,2

0 0 0 0 0 0

7~'71p:

2.2.1. Yielding at one qIth,:- element ends Assume the yielding of end k (k may be i or j , the plastic criterion takes the form f ( N k , M, k.M: k,ak) : {j%,~}r { F e } - N p ( 1 + a k ) = 0 .

(5)

The incremental element equilibrium is given in lunction of elastic phenomena: { dF,.} : ]K,.] • { dq':}

(6)

where { dq~} is the increment of displacements producing elastic deformations. The partition of deformations into elastic and plastic parts leads to the following expression: {dE.} = IK,,]- { dq,.} --[K,.I- { dq~,l}

(7)

where { dq/~} is the increment of displacements producing plastic deformations. Introducing the normality law as { dq{i} = d,~,,k • {.%}

(8)

where d/It, k is the plastic multiplier increment. The element forces can be written as { dF,.} = IK,.I- {d%} - [ K , . ] . {Jk/2} d4, a.

(9)

During the plastic flow, we have dfk = 0: d f k = { c)fk ~

J

dF,,} + (c)fk~dak=O.

kaa J

(lO)

The strain-hardening parameter L defines the proportionality between the increment of isotropic strain-hardening and the plastic multiplier increment. The increment of the strain-hardening dak is given by

A.M. Motuuned. M. Lemaire/Slructural Safe(v 17 (19951 167 193

: L da,,

)

173

(1 l)

Equations ( 9 ) - ( 1 1 ~ allow. us to describe the increment of plastic multiplier as

da,,, =

LN~, + t.'\"~}'

IK,.]-{:\,'~}

' { dq,.}.

(12)

The vector {.,~'~} is constant for each plane of the linearized plastic criterion. The quantity in parenthesis is constant as long as the strain-hardening parameter L is unchanged. So, we can calculate the total plastic deformations by simple integration of the above expression.

L,,V~, + {A,"~ }' [ K,. I ' {-'%'~ }

A/'k =

=

LN~ ÷ {.\'i}'

IK,,I. {,%}

" / dq,. }

" {'t,} + B

(13)

where B is the integration constant depending on initial conditions. At the origin of yielding (t = t*), the plastic deformations are nought. So, we have {q~} = {q~} = {q~}, which means that {q';}'={0},

AT,a = 0

and

ak=0

(14)

where the exponent ( ' ) means a quantity evaluated just at the begining of yielding (i.e. at t = t*). From Eqs. (13) and (14). we find B = -

{.\"~ }'.

I K,.I.

{q;}-

(15)

LN;, + {,~~},. IK,. I. {>,~} At this point, the plastic criterion is just verified (i.e. there is no strain-hardening yet):

.1: = {.\, }' {/:~7 } .... /v r = 0

(16a)

which leads to N,, =

{>,~ }' {/-~" } = {..\4o}"- I K e l {q,*,}.

From (15) and B =

(16b)

16!,we can write the constant B as Nr

(17)

LN~, + {.\", }'. I K,. I • { > 4 } By substituting Eqs. {17) and (13) in the integrated form of Eq. (9), we can find the expression of total element forces as {F,} = IK,,1' {q,.} + {F,,} where

(18)

A.M. Mohamed, M. Lemaire/Structural Safety 17 (1995) 167-193

174

'-P

.....

.M

Fig. 4. Integration of plastic displacements and initial conditions.

IK~.I = [K,,] -

[ X , l . {M}. {M}'-[t(,,1 LN 2 + {.K'~}'. l Kel. {.&~}" j X,, 1 •

{F"}=N'LN~, + {.&~}'. [K~.I. {;\'~}' We keep in mind that the elastic-plastic stiffness matrix [K,.] is the one which relates nodal forces to nodal displacements, while the vector of equivalent forces {T~} represents a constant quantity corresponding to a vanished displacement vector (i.e. initial conditions). Equation (18) is a generalized form of the Murotsu's formulation [9], with the extention to the case of strain-hardening mechanical systems. To find the perfectly elastic-plastic behaviour, it is sufficient to put L = 0. Figure 4 shows the element's elastic-plastic behaviour indicated by the points O, P, Q and M. For incremental methods (Eq. (6)), the analysis is carried out according to the bi-linear relationship OPM. For the integrated method (Eq. (18)), the behaviour is only assimilated to the linear path QM, the distance OQ being the initial condition. In the case of linearized plastic criterion, the tangent stiffness matrix [K~] is constant during each loading increment as long as the strain-hardening parameter is unchanged. At any yielding level, Eq. (18) gives a linear relationship between element forces and displacements, having the equivalent forces as initial condition. These equivalent forces are not functions of external loading, but only of internal strengths of yielded members. They vanish in case of pure elastic behaviour. In reliability context, these equivalent forces represent the contribution of strength random variables.

2.2.2. Yielding at both element ends Now, let us consider the case of two generalized plastic hinges at element ends. The plastic criteria are reached at both ends. A similar development leads to the following form of the elastic-plastic stiffness matrix: [K,,I = [K,.] - I K,.I . (g,[.,~;~,l - T,,j[A4, I - Tj,[.M,,] + T,,[AFj2] ) - [Re]

-

(19)

,rj,

and of the vector of equivalen! {r~,,} = N,, [K~I. ( ( Z ; -

T,j) {,X;} + (77,,- ~,) {N;})

(20)

A.M. Mohamed. M. Lemaire/Structural Safeo, 17 (1995) 167-193

tFe}, A

v

P .......

If •(

175

;

'

~,. •

Fig. 5. E l e m e n t b e h a v i o u r d u r i n g u n l o a d i n g .

with I,'%1 = {..<'i} - {A4}'. z , = {~'~}'. l x , , l .

{,<},

~ k = {.,%'~}' - [K,, ] . {.&~} +

LkN~,.

2.2.3. Cases of irregular behaviour Until now, the described model considers only the general form of elastic-plastic behaviour. But, it cannot take into account neither the cross-section unloading, nor the behaviour at criterion edges formed by intersections of planes. So, we should introduce a special treatment of such behaviour, without disturbing the model linearization.

Unloading of cross-sections. Even in the case of increasing external loading, local unloadings may occur due to redistribution of internal forces. This phenomenon is observed in offshore structures after buckling of bracing members. During yielding, the rate of plastic dissipation energy must be positive. A calculated negative value of this dissipation indicates the unloading of the cross-section and returns the behaviour to the elastic state. After detecting the cross-section unloading, we must keep in memory the reached level of plastic deformations. Let us consider the behaviour shown in Fig 5. The unloading at point P follows the initial stiffness matrix, but is shifted by the contribution of plastic deformations at the unloading point P. In the unloading phase, the total element displacements are given by {q,,} : {q;} + {qS'}*

(21)

where {q~,~}* is the vector of generalized plastic deformations, evaluated at the unloading point. Knowing that the element nodal forces are given by Hooke's law, expressed in function of elastic deformations only, the element equilibrium equation during unloading can be written in the following form:

{F,.} = [ ~ , l . {q,,} + {K,}"

with {T,.}* = - I

K,, I • {q[~ }'

(22)

A.M. Mohamed. M. Lemaire/Structural Sal~'(v 17 (1995) 167-193

176 w

where { F,, }* is the vector of equivalent forces in case of unloading. Note that this equation keeps the same general fonn of linearized equilibrium as given in the previous section (Eq. (18)). In order to reduce computer memory used when saving plastic deformation vectors, it is sufficient to save only plastic multipliers A~,: the plastic deformations can be deduced by the normality law.

Behaviour at singular points. The linearization of plastic criterion leads to a certain number of edges, known as "singular points", where the criterion derivatives are no more continuous. These singular points limit the loading increments and need special treatment. Let us consider a singular point P formed by n intersecting planes. We have n verified criteria relative to each plane equation: f, = {:\/'~}'. {F,. t f2 {:\L_}'. { F, t •

.

,t;,

LN~A,,~ L N~ A,,~ .

{AJ;,/". I ~; 1

-

L:~7,4,,~

mp = 0 ] Np 0 .

x,,

.

(

2

3

)

0

Using Hill's principle, the normality law takes a generalized form [ 1 4 ] { dq~;} = dA,,, {.&~ } + dA,,2 {,,%'~} + . . . + dAp,, {,,\";,}.

(24)

During plastic flow. we should have d fl = df2 . . . . . d f , = 0. Hooke's law, the partition of deformations and the generalized normality law allow us to establish the following relationship: I G I - { d A } = I~Vl'- { dq,.}

(25)

with

i GI __ { g,, -- {.\:~}' l K,,I • {A(,}: g, = {.Q}T [K,,I "{J~} + L N ] '

dA}': E

igj, i=j,

da,,, da,,,,].

ixl'--[{.v,~' {,<}' { m }' = {.< }'. I x,, l

{,v~},!.

where [ G] is the matrix of criterion derivatives; { dA} is the vector of plastic multiplier increments; {q,,} is the vector of nodal displacements. The solution of this equation is possible as long as [G] is not singular. The increments of plastic multipliers are given by { d A } = I GI ' '

l.Vl"{dq~}.

(26)

Knowing that [G] and [X] are constant, the integrated form of the above equation gives:

{.,,\ } = I(31 -'-I,'cl'.

{q,,} + {t3}

(27)

where {~} is the vector of integration constants. Hence, the element equilibrium equation takes the following form: {F,,} = IK,,1. {q,,} + {if,,}

(28)

A.M. Mohanu,d, M. Lematre/Slructural Se{/Fty l 7 (1995) 167-193

Plastic flow at the same singular point Plastic flow leaving - - k,,~{~e} @ the singular point

m/ Plastic criterion after strain-hardenin

Plastic criterion at initial position

177

I --I

iiii:

~'~~~i~lO~oad| -

n

" ]lunloadigng

C Total unloadingo~ {~Pl the cross-section • Fig 6. Possible kinds oI behaviour at singular points. with [K,.I = [ K , I - [ : t ' l . l G ]

'.l)dl'

and

{U,}=-IA'I.{/3}.

This expression gives the general form of the element behaviour at any singular point.

When reaching a singular point, the behaviour must follow the general formulation (Eq. (28)). But the evolution from a singular point depends on the whole structural behaviour. As we deal only with finite loading increments, the behaviour tendency must be checked. It may take one of four possibilities (see Fig. 6): ( 1 ) Plastic flow at the same singular point. (2) Plastic flow leaving the singular point. (3) Partial unloading of the cross-section. (4) Total unloading of the cross-section. In order to find the appropriate evolution, tests on criterion and plastic multipliers are introduced at the end of each loading increment. If checks do not approve the compatibility of behaviour, the structure state is modified and the increment analysis is recalculated. A very low number of iterations allow to determine the compatible state of equilibrium. Evolution f r o m a s i n g u l a r point.

U p d a t i n g ~/'strain-hardenin~ parameter. At several increment analysis, it may be interesting to update the strain-hardening parameter, especially when it represents buckling behaviour. The model must memorize loading history and separate random variable effects. Figure 7 shows element behaviour defined by two different strain-hardening parameters corresponding to elastic-plastic stiffnesses [Kq ] and [K. 2 ]. In this case, the vector of equivalent forces is given by:

178

A.M. Mohamed, M. Lemaire/Structural Safety 17 (1995) 167-193

~Fe~b ~Fe [K'e]

{qe}

{%} Fig. 7. Updating of strain-hardening parameter.

{Te2,, } : (Fe2) + ({F,'2 } L, N p - [ K e ~ ] {~1 })At: ,

(29)

where{Fe211 } is the vector of equivalent forces corresponding to phase 2 after yielding at phase 1; [Ke2], {Fe2} are respectively classical stiffness matrix and vector of equivalent forces at phase 2; LI, {Aft } are respectively the strain-hardening parameter and the normal vector at phase 1; Apl is the plastic multiplier at the end of phase 1.

2.3. Negative strain-hardening In order to overcome the difficulty due to nonlinear behaviour during buckling, Melchers and Tang [15] used a very simple model for trussed structures known as elastic-residual. The method assumes that the behaviour of the member is perfectly elastic up to the failure load where some part, but not all, of the initial failure strength capacity is retained. As a matter of fact, the approximation in the post-buckling range is generally insufficient and leads to very low precision, which may have a significant effect on reliability estimation. Ambiguity arises from the fact that incremental analysis becomes discontinuous and does not meet real behaviour. For truss members, Moan et al. [ 16] and Wu et al. [17] used an improved axial force-displacement curve by introducing a negative stiffness matrix. Critical force and buckling curves are given by standard codes of practice for hinged columns. In the present study, we generalize this approach to beam-column members. In this case, standard codes of practice cannot give an accurate description of buckling properties, because of the effect of boundary rigidities and bending moments, and of the redistribution of member internal forces during the post-critical range. This means that the replacement of the member by a certain axial stiffness is not sufficient to resolve beam-column problems. The elastic-plastic buckling is characterized by a reduction of the ultimate axial strength and by the decreasing of the member capacity during the post-critical range. In the following section, we show how both of these two characteristics are incorporated in the mechanical model.

2.3.1. Reduction of ultimate capaciO, For beam-column members, the critical force is, generally, less than the squash load. This fact is taken into account by introducing a reduction factor ae to the axial plastic capacity Np. This factor is introduced only at the compression phase of the criterion (Fig. 8).

A.M. Mohamed. M. Lemaire/Structural SaJety 17 (1995) 167-193

179

Am ( oll]plt2551Oll/ ~ ~ -I / l-c~

\+I

\\

Fig. S. Reductionof the compression strength of the member.

C°mpressi°nIAm

I~

+1 ~

-a

~|' ~|~N~ a(ctNp)

sion.j~ ~

0
J~

>,-

Elementshortening

Fig. 9. Elastic-plastic buckling modelling by negative strain-hardening. In this case, the compression strength N:, is substituted by c~N~, (with 0 < oe < 1), while the tension strength is unchanged (equal to N , ) . For compression members, the plastic criterion takes the following form:

f,(N.M,a,)

=

J , CSign( M, ) ,ff-gSign( M: ) ,7,,Sign(N) M, + M. + - - N M :,, M :,: teN:,

(1 + a , ) .

(30)

2.3.2. Li,earization o./post-criticalbehaviour In order to take into account the post-buckling behaviour of frame members, we [13] represent the increasing loss of axial force capacity by a negative strain-hardening introduced in the elasticplastic model. Contrary to the case of truss members, the loss of beam member capacity should be accompanied by the development of residual bending moments. The strain-hardening parameter L allows the reduction of the size of the plastic criterion (Fig. 9), leading to a corresponding loss of member capacity. This parameter can be constant or updated at loading increments. The values of o~ and L are functions of end moments, of initial imperfections and of boundary conditions. The value of the strain-hardening parameter depends on the ratio of dominant internal forces. In case of member subject only to bending moment, we have oe = 1 and L = 0 (no buckling). In the case of truss member, we have ce = r r : E l / N , I= and L < O. But in the general case, a powerful identification method is primordial for the analysis of real structures. 2.4. Identification ~?/the strait>hardening parameter [13/ The main difficulty of buckling analysis is to determine "accurately" the elastic-plastic buckling curve of members subiected to different loading modes and boundary conditions. During buckling, the deformed shape of a member must satisfy the internal and external forces equilibrium, as well as boundary conditions ( Fig. 10).

A.:'V/ Mohamed, M. l, ematre/.Structural Sqfetr 17 (1995) 167-193

180

P Zone of compression yieldin Zone of compression yielding

/ %

~

/ _

\ Icldl ng

[ ~

~% " Zone of compression yielding

+p Fig. 10. Buckled m e m b e r shapes.

In order to find the so-called "'exact" member shape, a numerical method was elaborated which is based on a shape function (m finite element context) formed by a high degree polynomial. The coefficients of this polynomial are those which verify the boundary conditions and the equilibrium of internal and external forces at several cross-sections distributed over the element length. The equilibrium in the rest of element cross-sections is approximated by interpolation. With reference to chtssical nonlinear methods 118 201, our model allows a more satisfactory representation of the curvature (for the same discretization). A little number of intermediate crosssections gives a very good precision. Also, it has the advantage of evaluating any equilibrium position without needing to carry, out an incremental analysis. Knowing the elastic-plastic deformed shape, the element axial shortening is obtained by the sum of axial strains, and of relative end displacements due to curvature. The calculation of a few points on the buckling curve allows to evaluate the corresponding negative strain-hardening parameter. This is introduced to the elastic-plastic model in order to perform linearized mechanical analysis. To illustrate the proposed method, let us consider the element shown in Fig. 11. At any crosssection, the equilibrium must be established between internal m o m e n t / v 4 ( x ) and external moment MI ( x ) .'~(.r) = Mr(.v)

=~-

El{ N)A'(x) = ,.~/'vtx) ~- M(.v)

(31)

where ,t.'(.r), y ( x ) are respectively element curvature and deflection at x; El (N) is the elastic-plastic flexure rigidity (function of axial force N) and M(x) is the applied bending moment. In expression (31), the deflected shape is not, a priori, known. Hence, we consider a certain number of cross-sections tl + 1 distributed on the element length. Noting n~ as the number of boundary conditions, the polynomial degree t~ is given by' n = n, + n,,. y ( x ) = ~,,~ + c~l.v + ~:~.v-' ~ -. 4- a,,_v"

(32)

where x is the abscissa on the element longitudinal axis. The coefficients o~,, (i = 0 . . . . . n) are to be determined. Rotation and curvature are obtained by the derivatives of the deflection polynomial:

A.M. Mohamed, M. Lemaire/Snm'tutzd SqI~'U 17 (1995) 167-193

181

t:ig. 1. Compression clement conliguration. 33)

34)

For bean>column members, the cross-section behaviour may be elastic or elastic-plastic. The internal compatibility condition describes element curvature A' by [21,22]:

35) where ]z is the cross-section height, e~ and e, are maxinmm and minimum axial deformations, crr is the yield stress and H is the Young's modulus. An iterative process, allovrs us to find the polynomial coefficients which satisfy, besides the boundary conditions, the following nonlinear set of equations:

(36)

where ,'~( N, 2t'~) is thc internal moment at the cross-section k (function of the normal force N and the curvature ,.t.),): M! .~;) * N v(.v~) is the external moment composed of the applied moment and of the contribution of axial force. 2.4./. I n c o r p o r a l i o , ill t/l~" /m~'ari:ed mode/ The proposed mechanical model allows the elastic-plastic analysis of structures in a piecewise lmearized form. In each loading increment, the whole rigidity of the structure is maintained constant. The supcrposition principle remains valid in each increment. Therefore, we can divide the structural system into two sub-structures (see Fig. 12). The first represents the structure without the critical member. The second represents the isolated member having for boundary conditions the stiffness of the rest of the structure. For this isolated member, we can perform buckling analysis by resolving the equation set (36). This procedure satisfies displacement compatibility, as well as forces equilibrium between isolated member and the remaining structure.

182

A.M. Mohamed, M. Lemaire/Structural Safety 17 (1995) 167-193

÷L,

X Real Structure

Structure without critical member

Critical

member

Fig. 12. Decompositionof the structural system.

N X.. P~ Ot

~iemenl a×ml sh~r~enmg

O

qij

p-

Fig. 13. [demificationof parameters from the linearized curve.

It is to be noted that the proposed method allows us to calculate any point of the buckling curve without carrying out incremental analysis ; the calculation time is considerably reduced. The parameter o~ is calculated by the ratio o~ = N~:,/Np. Noting that to obtain a linearized form of the buckling curve, it is sufficient to calculate a few equilibrium points at reasonable intervals of axial shortening (Fig. 13). A linear segment joins every two successive points. For each segment, the negative strain-hardening parameter can be evaluated in function of the slope AN/gqi;. These values are injected in the elastic-plastic model in order to take into account the buckling effects.

3. Formulation of safety margins The evaluation of failure probabilities of components and systems implies the knowledge of the safety margins, which may be linear or nonlinear. The advantage of using linear safety margins lies in the high degree of efficiency. In the following, the safety margins are given with reference to the elaborated mechanical model. The safety margin represents a random variable with distribution defining the allowable space for the loading to produce failure. The safety margins can be divided into two categories: the first called resistance safety margin as it is related to the ultimate limit state of components, and the second called seta, iceability safet3, margin as it is related to the serviceability limit state.

4. M. ,~'lohanled, M. Lematre/Slructural Sa/ety 17 (1995) 167-193

183

3.1. Resistance safety matin'ins

The resistance safety margins are used to define the component surviving conditions. Knowing that linearized plastic criterion defines the failure condition of cross-sections (structural components), the safety margin M, is simply obtained by inverting the signs of this criterion: M , = N~,

([,CSi~,,n( M, )

×,,

. M,

~'7:CSign( M_ I

z,,~

- M . - J , Sign( N ) N

(37)

where Zp,, Z~,: are plastic coefficients giving dimensional characteristics of cross-sections: Zr,>, = MI,,/N~,, Zj,: = Mr,:/,,\',. This safety margin is a linear function of internal forces and cross-section capacities. 3.2. Displacement

sa./i'tvmat:~ins

As a matter of facl, the strength failure of the structure does not represent a satisfactory criterion to define structural failure. For redundant structures, the mechanism formation occurs at a very high level of deterioration at which the structure is not useful any more. In other words, a resistant structure which cannot ensure acceptable functioning should be considered as a "failed one". The serviceability limit state is defined by allowable values of working properties (displacements, crack widths, etc.). these allowable values do not depend on the structure itself, but on its function. For this reason, additional safety margins are introduced to take account of allowable displacements. Contrary to resistance margins, the displacement safety margins are not related to cross-sections, but to structure nodes at which displacement should be controlled. These margins are given by: M,~, =

q[

q, ~ ii

(38)

where M,~, is the displacement safety margin related to the degree-of-freedom number i; q~' is the allowable displacement: q, is the displacement produced by the external loadings and ?]i is the displacement due to yielded member capacities.

3.3. Calctdation (~/h!/hwncc ,:'oe~cients In the proposed mechanical model, the tangent stiffness matrix is constant between two successive deteriorated states of the structure. The nonlinear analysis is then reduced to a finite number of linear sequences. The integration of plastic deformations is introduced in order to distinguish between the contributions of different random variables: this leads to the element equilibrium equation (18):

{,~5} = IK, I. {q, ~ - ~ F , /

(39)

where [K,. I and {F,. } remain constant during each linear sequence of the analysis. The assemblage of this system leads to the equilibrium of the whole structure: IKI.{q}=IF}

{F}

(40)

where { F } and {F} are xcctors of external forces and equivalent forces assembled in the global coordinates. Tile vector ~f:} is divided into a certain number of groups of independent loadings

.4..'14 Mohanted, M. Lenmire/Structural Sqlety 17 (1995) 167-193

I g4

";;,g,b'" A group is a set of loads perfectly correlated. Similarly. the vector {F} is divided into a certain number of equivalent forces "ne f" (equal to the number of yielded elements). Each vector of equivalent forces is related to the strength in pure normal force N:,. The above equation can be written as follows: /,a'[,

;,',' [

l=]

.,=l

(41)

with i

{b;} = {A;}P,,

and

{F;} = {A:}N;,

where {A,}. {A: } are ~ectors of deterministic coefficients. P,, are the loading random variables and N,: are the strength random variables. Knowing that the mechanical model allows us to obtain the contribution of each random variable in a piecewise linearized form. the output of this model is a linear function o f its input, which is written as •

~-£~',,'\,':,

N=~c,,,P, i=l

(42a)

::1

:lk't~

M, = ~

c,,,,,P,,

-

-

Z

nk'];

M: = ~ c,,:,P,

~:"',;' '\'`'`

(42b)

!:-I

I=l

,:, r

\ ~ i::,,~ N

i-I

(42c)

r:=

where c,,;, (,,;, c,,,. c,,,,. c,,,_,. C,,:; are the coefficients of linear functions (they are evaluated by the mechanical behaviour). By' substituting Eq. (42) in (37). we can write the general form of safety margins as follows: .

,¢1,~ = N,. a + ~ ' ,=~

ii. + - + " Z;,k Z;,,

N;,,

-~ ,=1

a,+ ~' + Z;,,~

Pr,

(43)

The coefficients a,. h . b_, a , h,;, l?:; are respectively the influence coefficients due to external loading P,., and to cross-sectional strengths N:,. The dimensional variables ZI,,.~, Zt,:k may be considered as random, which leads to a nonlinear safety margin. However, this does not alter the linearization process of mechanical behaviour. The elaborated programme allows the calculation of these coefficients for the case of plane and space structures.

.?.4. Sirra'tufa/ reliahil:tv Knowing the satet\ margin (expression ( 4 3 ) ) . the failure probability can be evaluated by the --) ..) well-known relationship [ , 3 . , 4 1 . .

P,

=

P( ,,'d,~ <. O~ -- ~1)( --13H; ~ )

(44)

A.M. Mo/lamed, M. Lemaire/5"truclural Sq/i't.v / 7 (1995) 167-193

?';tructurc

I ~mdmgs

N:

185

Reliability

,nOue.ce

....-""~ECHANICAL MODULE~'"~

/'~

coefficients

,:l

3",I Mechanical Results

Reliability Results

Fig. 14. Scheme of mechanical anti rcliability analysis.

where qb(.) is the standard probability distribution function and ~HLI. iS the Hasofer and Lind reliability index I251 evaluated for the cross-section "k". At structure undamaged state, the influence coefficients due to external loadings ai, b.,.,bz~ are evaluated by the linearized mechanical analysis. By evaluating the safety margins (Eq. ( 4 3 ) ) , the cross-section failure probabilities can be calculated (Eq. (44)). Then, failure conditions are introduced successively at critical cross-sections in order to form the most probable damaged configurations. With the branch-and-bound method [9], dominant failure modes can be generated. The failure probability of each mode is obtained by forming a parallel system of basic components. The structural reliability is estimated by' modelling the identified significant modes as components in series system. The global failure probability is given by:

(45)

where M,il is the conditional safety margin of the failure path "i" at the level "j" ; n is the number of dominant failure paths; mi is the number of basic components in the path "i". In order to identify dominant failure paths, methods like the branch-and-bound method can be used, but also the /3un,,ippinv method 1261. Even for a limited number of failure modes, the "exact" calculation of system failure probability is very difficult. The Ditlevsen bounds [27] allow a reasonable estimation of the system reliability. The scheme of structural analys~s programme is shown in Fig. 14. The structural and loading mean characteristics are introduced in the mechanical model. This later identifies the strain-hardening parameters for critical members. At each structural state, the model gives the piecewise linearized mechanical response as well as the influence coefficients for internal and external random variables. These coefficients are associated with random variable characteristics. By mean of classical reliability procedures, the failure probabilities of components and systems can be calculated.

186

,4.M. Mohamed, M. Lemaire/Structural Sq]ety l 7 (1995) 167-193

4. Numerical examples The proposed model is applied to the estimation of offshore structure reliability. The considered examples represent classical plane and space offshore frames. The very good precision of the linearized mechanical model is verified in all numerical applications. According to usual practice in offshore engineering field [28,29], the reliability analysis of the structures is carried out for a normally distributed design loading with a coefficient of variation equal to 40% and log-normal strengths of members with a coefficient of variation equal to 8%. Correlation is considered between member strengths; load effects are fully correlated, but no correlation is considered between loads and strengths. The software S T R U R E L [301 has been used to calculate component and system reliabilities. 4. I. Tubular plane f r a m e

The overall geometry of a two-storey braced frame is represented in Fig. 15(left). The structure, studied in a previous work [31]+ is loaded by a single horizontal force applied at the top-level. The mean value of the load is taken as the design load given by the API RP2A. The frame is modelled by 23 tubular elements using a linearized plastic criterion with parameters y = 0.25, ( = 7/= 0.00. Figure 15(right) shows a good agreement between the results given by the

100

90 J

kit0

®

@

'?

is0

,~+.

30

,,_

A

10 r 0

,~ 2

I 4

I 6

I $

L 10

[ 12

14

16

Lateral displacement (cm)

Fig. 15. Plane tubular frame: Platlorm configuration and mechanical response.

18

A.M. Mohamed, M. Lemaire/Structural Safi, ty 17 (1995) 107-193

187

ii s8

S7

S5

/14

EI~

S6

S3

S4

S1

,S 2

c.;

P@ P@

,,@

Sn Failure of section "n" E n Failure of element "n" E R Reaching of residual capacity A

Displacement safety margin

F@

,,@

~-~-TJ' Reliability index

Fig. 16. Partial failure tree defined by dominant paths.

present model and those of other numerical and experimental studies [ 31 ]. The loss of frame capacity is observed after the failure of the bracing members number "14" and "15". The buckling of member "14" is very well represented by the negative strain-hardening and it can be observed that results are closer to the real behaviour than those of the nonlinear analysis given by the Steel Construction Institute 1311. Under design load, the reliability analysis is performed. Figure 16 shows a partial failure tree indicating the dominant failure paths. At the undamaged state, the critical members are found to be those forming the upper bracing system (members "14" and "15"). The global reliability of the structure is estimated by /3,,, = 1.60 (i.e. Pt= 5.4 × 10 2). This value corresponds to the buckling of compression member. At damaged state, the branching begins from critical failure paths corresponding to the buckling of member "14" and to the tension failure of member "15". At the structure deteriorated state (failure of upper bracing system at lm,el 4), the reliability of the structure is evaluated as : /3,>. = 1.73 (i.e. Pt = 4.0 × 10 2). At this level, the reliability index is related to two successive failures (members "14" and "15").

188

4.M. Mohamed. M. Lemaire/5"tructural .Sqfe(v 17 (1995) 167-193

4.2. 7"td)ltlar Sl),Ce ,/}'a,~e

The studied structure is shown in Fig. 17(left) [321. It consists of a three-storey space tubular frame supporting an equivalent storm loading. The mechanical response is shown in Fig. 17(right) by the relationship between loading factor and lateral top displacement in the direction of the y-axis (node "'42"). Our results present a pleasing agreement with the nonlinear analysis performed by the program USFOS [331. The mechanical response is characterized by the buckling of lower bracing member number "57". Under storm loading, the structure reliability at initial state is found to be:

/3,~, = 2.66

{failure probability Pt ~ 7.00 × 10-2).

For the undamaged structure configuration, Fig. 18(left) shows the reliability indexes of critical components ( i.e. member ends). Between them, the most critical failure modes are yielding at sections $20 (/3 = 2.70) and Sis (/3 = 2.91 ), and the buckling of the lower bracing member Esv (/3 = 3.91). The partial failure tree considering the most critical failure paths is shown in Fig. 18(right). Under the design loading conditions, the reliability of the structural system at different levels of damage is given in Table 2.

Z ~

"< 16,00m_>" 41

42

X Z6

7.4 7.2 Z0

{

1.o 0.8

o.6 OA 0,2

Displacement at node "42" in the "y" direction

Fig. 17. Platform conliguration and mechanical response.

1.M. Mo/~amed. M. Lemair~'/Slrucmral Sa/i,n 17 (1995) 167 193 Table 2 Structural .~ystem reliabilit~ at dil]crent Level Reliability index 1 /3~,, = 2.66 2 /3,,, = 2.87 3 ,{3,,, 3.7 ~) 4 /3,,, = 3.94 :

189

damage levels Most probable damage Yielding at $20 Yielding at b'js Buckling of mcmbel /:~Failure of member 1:~,

En Failure of element "n" ]•

r'e'R',t7~

I

Level I

*I ~~ ',~/" ~ /

h~_.~

,,~,~,:. pc \

I'~'R't~l

'//¢ ~ - ~ A ' su'." { _ ~ 7 i

Level 2

Level 3

Level 4

Fig. IN Reliability aml vsi~ <~t the plaU~rm. ~z ) So, ram rehahlit> indexes at initial state. (b) Partial failure tree. The herein model represents a cotnplete and improved approach of simplified methods, as for example, which considers a single component with a resistance and load effect referring to the base shear 1341. 4.3. Compuri.son ~1 n~o bru¢'ing ~vstem.~" In oil"shore engmeerit~g, two types of bracing , y s t e m s K-system and the X-system. The choice of the convenient of optimization. The advantage of the reliability method based on mechanical as well as reliability considerations. parameters: cost and rcliabilit>.

are widely used; they are known as the system represents an interesting p r o b l e m lies in its capacity to m a k e c o m p a r i s o n This introduces the design fundamental

4.3./. Compori.~on wi:/1 identicul cross-sectio,s In an elementary study of this problem, two braced plane frames, representing an intermediate panel of platforms, are considered (see Fig. 19). The two frames are nominally equivalent (identical cross-sections and global dimensions t. Two horizontal loads are applied at the top level.

190

A.M. Mokamed, M. Lemaire/3lru('tural Safely 17 (1995) 167-193

~,P

).

2

':3"

"~:"

(7" "..J

4

~P/2

<¢ #

~,P

~.P/2

,9,///////

5-

::g

:V

l

////

Y/:;

I

.< 6,~0,,, > <

~:Om ~,.

<

6,7{M1~

<

-6'7°" >i

Fig. Iq. Tx~o bracingsystems.

2,0

.E 1,2 = 0,8

0,4

th)rizontal displacement (cm) 2

4

6

8

10

12

Fig. 20. Mechamcal response of bracing systems.

The dimensions of these bracing systems are given by the society Elf Aquitaine Production, according to experience in platform design methods. It is seen that the length of bracing members is equal to 20 meters for the K-system, and to 30 meters for the X-system. For identical cross-sections, the X-bracing has 50c~ steel excess with reference to the K-bracing. Noting that cost excess will be higher than 50% because of additional weldings at joints 1 and 5. The question to be answered is thus: which one of the two systems has a better ratio: "'cost/reliability" '~ Figure 20 shows the mechanical response of both structures. It is observed that the X-bracing is more rigid and has a higher ultimate capacity. Also, the post-critical phase is sudden in the case of the K-system. In both cases, the negative strain-hardening approach gives good results. For the same computer, a comparison of calculation cost shows that our model is about 60 times faster than nonlinear models [ 181. In both frames, the failure can be identified by the loss of ultimate capacity, which is observed only after the bracing system failures. For the K-bracing, its failure is reached either by buckling of nlember "3" or by yielding of member " 64" followed by buckling of member "3". In the case of X-bracing, the system failure implies the failure of both bracing members. The failure probabilities are given by:

AM. Mohamed. M. Lematre/Struclural Sq/eO' 17 (1995) 167-193

191

Table 3 Comparison of bracing systems ,x'ith identical cross-sectional area. Propeny K-bracing system X-bracingsystem Frame ultimate capacity (.4P),.~ = 1610 kN (AP)~.~= 2054 kN Bracing reliability index /3,,, = 1.30 /3,,, = 2.23 Bracing steel weight W= 1206 kg W = 1809 kg Table 4 Comparison of bracing systems with identical ultimate load. Propeny K bracing system X-bracingsystem Frame ultimate capacity CAP),, = 1610kN (AP),~= 1610kN Bracing reliability index /3,,, = 1.30 /3~, = 1.35 Bracing steel weight PC= 12()6 kg W = 1420 kg

,,,,,,,,, = P

::

(-'/IM,,,

<

Ui

M, _<_0t

< 0)

J]

.

The system reliability analysis shows that the buckling of the compression bracing m e m b e r unloads the tension member in the case of K-bracing, while it overloads the tension m e m b e r for the X-bracing. As a matter of fact, the X-bracing is more reliable which implies greater steel cost, compared to K-bracing, see Table 3. 4.3.2. Compariso~z with identical ultimate load

The choice of one of the two systems is not simple, because of multivariation of design parameters. In order to make comparison possible, we should retain some of structure variables and see what happens to the rest. Thus, we can compare the two systems for an identical strength reliability indexes, fabrication cost, ultimate loading capacity or serviceability reliability indexes. In our study, we choose to reduce the bracing cross-sections (maintaining the same aspect ratio) of the X-system in order to obtain the same mean ultimate load as for the K-system. For this configuration, the X-bracing has 18% of exceeded steel quantity with a reliability similar to the K-bracing (see Table 4). This means that the K-bracing has a better ratio cost/reliability, but its sudden failure is only dependent on the compression bracing member. On the other hand, the X-bracing has the advantage of keeping a residual resistance e~en atier the buckling of the compression member, which may give "an alarm ring" to allow reparations and aids. The confusion of the better choice refers to constructor's political decisions, which may explain why both systems are equally used in offshore constructions.

5. Conclusion A linearized incremental method has been presented, which allows an efficient and accurate reliability analysis. The mechanical model takes into account the interaction between internal forces, as well as elastic-plastic buckling. Cross-section unloading, behaviour at criterion singular points and updating of strain-hardening parameters are taken into account. A method allowing the identification of negative strain-hardening, equivalent to elastic-plastic buckling, is proposed. The integration of plastic defonnations allows us to separate the influence of different internal and external random

192

A.M. Mohamed. M. Lcmaire/Structural SqlFty 17 1995) 167-193

variables. As the mechanical model describes linear safety margtns, the reliability analysis is simply done by classical procedures. Numerical examples have been presented in order to show the performance of the model. With reference to nonlinear mechanical models, our method shows a very good efficiency and a satisfactory precision. With reference to classical nonlinear reliability analysis, out model needs very little computation cost. So, the proposed method allows to take into account a realistic mechanical behaviour even with structures composed of a very large number of members. Compared to most of the other available techniques, this method is a way to perform an accurate reliability analysis of structural systems with a reasonable computational cost.

Acknowledgements The authors wish to thank and acknowledgc ELF AQUITAINE PRODUCTION for their financial support of this research project. References 1 ] Madsen H.()., First Order ~s. Second Order Reliability Analysis ol Series Structures, Structural Sa,flety, 2 (1985) 207-214. 21 Hohenbichler M., (]ollwitzer S., Krtlse W. and Rackwitz R., New Light on First- and Second-Order Reliability Methods. Structura/Sq/~'>. 4 { 1987) 267 284. 3 ] Murotsu Y., Okada H., Ni\~a K and Niwa S.. Reliabtlily Analysis of Truss Structures using Matrix Methods, J. Mech. l)esig~. 102 { 1980}. 41 Gu6nard Y., Application ot System Reliability Analysis to Offshore Structures, The John A. Blume Earthquake ReseaFch Center, Report No. 71. Stanford University, 1984. 5] Moses F. and Stahl B.. Reliability Analysis Format For Offshore Structures, Proc. lOth Annual Ojf~hore Technology Cot!t?, Paper No. 3046, Houston, 1978. 6] Schmidt L.C., Morgan RR. and Clarkson J.A., Space Trusses Analysis with Brittle-Type Strut Buckling, J. Structural Div.. ASCE. 102 (1976) 1479 1492. 7] Sorensen J.. PRADAS - Program for Reliability and 1)eslgn of Structural Systems, Structural Reliability, Paper No. 36, Institute of Building Technology and Structural Engineenng, Aalborg University, Denmark, 1987. 8] Karamchandani, A.K. and Cornell C.A.. Reliability o1 Trusses Subjcct to Non-Proportional Loads, Proc. 6th Int. Cot!fi (m Alqdicali(,ns q[' Sta:i.~tic,~ and Probahilit~ in Civi/ t:ngineermg, Mexico City, Mexico, 1991, pp. 377-384. 91 Thoft-Chrislensen P and Mtu>tsu Y., Apldicatio, (!/Structural Systems Reliability Theol3,, Springer, Berlin, 1986. [10] Eli Aquitaine Production. EI,FIAB Program ot Reliability Analysis of Offshore Structures, Ell Aquitain Company, Pau. France. 1990. II] ARP[:,J, Program ul F,eliabilit 5 Analysis of Trus~,cd ,rod Framed Structures. CTICM, Ell Aquitaine Production and Ifremet, Paris. France. 190 I 121 Lemailre J. and ('haboche J.L.. M~;conlque de,s Al~itcri~eu.~ 5olides. [!dition Dunod, Paris, 1985. 131 Mohamcd A.M., Modclc MOcanoliabiliste Lindaris0 pour l'Analyse des Structures. Applications aux Plates-Formes Marines, Ph.D. Thesis,, Blaise Pascal University. ('lermont-Femmd, France, 1993. 141 Koiter W.T., Stres>-Strain Relations. Uniqueness and Variational Theorems for Elastic-Plastic Materials with a Singular Yield Surface, Quart. AiJpI. Math.. 11 /3) (1953) 330 354. 15] Melchers R.E. and Tang I..K., Reliability of Structural Systems with Stochastically Dominant Modes, Monash Universits,, Australia, Civil Engineering Report N o . .~, 1983. 16l Moan. J., Amdahl J.. Engseth A.G. and Granli T, Collapse Behaviour of Trusswork Platforms, Proc. 4th Int. Conf o , l:Pehaviotu qf g![l:sh~,e Structures, Amsterdam, 19S5, ]171 Wu Y.I,. and Moan T.. A Slructural System Reliability Analysis of Jacket, Ptvc. 5th Int. Conf on Structural Safety a , d Reliability, 1989

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193

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