PlIoceedingso! the First (1991) Interhation'a,f,(})!!short! andPolat Engineering 'Conjel'enceEdinbur-gh, ..Vhited'1(ingdom,.11.16 A,ugustJ991Copyright1991 by P'helntetnationalSoeietyo:f!OJfShote 'andPolat'EttgineetsISlffl,rO~9'6261'(J4~5-3 ($et); ISBN O"96261(J49~6(Vol/V)

REASSESSMltN'l" OFO"FSHORlt STRRLST1tUC'VUlt:ES~ $tM):AKJFjJj(i)'WN\~N.CYCLIC NONtlNEkR FEM .ANAtS'ES'

Oyvind llellanand BjotnSkallerud:SINTEF Structural Engineering

TrondheilIi, Norway

Jotgen AmdahlandiTotgeirMoanDiy. of Marine Structures,.Norwegian Inst. of,1'ecbnolog,y

Trondheim, Norway

INTROJ3lUC'1']ON

Offshpre strttctures ai'S' g'enerally desig'nedaccording to the Olt:LIfiate Limit State Ct.JtSj, theFatigue Limit State fFLS) and theServiceahHity LimitState (SLS)eritE!:ria. The ctlnventional uts chMk isbased on linear elastrid analyses. BachmeIliber ischecked wi:t:hrespectto yi'eldiJ'flJ, buckling etc. , forenvirClnmen,ta:1 loads with, a rettirnperiod of typically100 year, and for SPElC if ied funct1anal loads.

Some codes (ECeS; 1~78 and '.Vhet Nor'l'teg':i/an Petro~lEiurti Directorate, 1984) haVe introdtided d'esign cr i-terier to ensure surviv'a.f oftl:1e strticture in case of.accldental loads of abnormal sttEiI1g'.th,typicallywitha return period of 10 OOl): years. These checks a':reca:rried' out in thee limit state of progressive Collapse(PLS). In the PLScOl'lboI, eli!lsbol,Hast.jjd behaviour isaOMp,ted as long a'S'" the overran integrity of thestructure is. mainta:il1ed.

!JJhe rela:tiol'fShip between OtS and PLS analyses isillustra:ted in 11'1g'\1':re 1, rot wave lClading.. Instead ofapplying strictly the 10 000 year wave load, it hasbeen common p,ractice to increment the TOO year waveforces proportionally up tOdol1apse of the structure.'.Phis is partly becaUSe the reserve strength as suchhas received considerable interest, and partly becausethe 10 000 year wave load selddm governs the deSign.S1.1ch an analysis is !!jenerally called a static pushoveranalysis.

Collapse analyses of the whole structure havedocumented a significant reSEi:rve strength, with systemcollapSe odcuring,a load's often 50% above the loadthat oause first (lIember failure. The question has beenraised as to whether these reserves can be of anybenefit ~ and howll!tich of the rEiserVeS can beutilized.

The design conditions for a platform may changeduring the serVice life due to

Increased tops'ide (6pe~at:idj'fal) loading Extenlfecf life beyond the,l'es:'ig'h service I He

34

Re~apprais.a{ of 'the enV'lll:ro I1ItH,nrtai!, h'5!:td'iitlg Reduded lClad~dariJyi:ng, capat:l:Ety due 'be d1I1iag:es"

cl1acks or oorroS::6.o

The donsaquenc:e of thesee.v.~Wffs:;mar be that {'j);t:st.'member failuraC5~di.lrS foral}" erlVd:iJfohitietiu'al litia'd)ShtaI1er than the reviSEld'des'i'gll l!,oatt;'; t:lf till!',! ULSdesig'l1 cdteriol1' is' to, be IIi!:!"\:; ,~'I:J, any tj'iml!fduring,oper'at ion i!n the Clofi'V'entlidna,lw1'ily', thE!' strUtlt'Ui:Ef' may'e,ither have to bEt s;trenlJllehecr~, 6r thef'Ufftft,'fonal.:loads must be redttced:. This Ilia'y fje' a' ",e'f:' SiX'pehSjjJ:ve ltask. On the o'the!' hand", i,t i,s,'unl:j}ke1i that 'bhe~structure will collapse' f:or a' s'incg'Ie' d~rs:tlgnwllve'Clct ton, dUE;! to the iirlherent, sbreflg''tihreseltvElsthat danbe denlonstrated.

In 1988, Shell pUbl ishEfd; a pl'og:tess'ive ptiocedur'l;lfor re~asSeSsIlie:nt (S'\:swarte'tial\ rg;8lf) ot'4,:liJ,z1.:ttt;l, non""linear statiC! pushdv'e~ anal'y,sl!'S'for.lftn\idt:l:i,r~'Sw11.08'1:1intelJritYdCluld not be dSnlQnsttra,tced' us':img: cOl1evenf"iional1inea,t m.ethbaS. This pape,rstHj,9't!'s:ts',l1.6Vttlt'i:!se more'cdmlllonpushdVe:l:i anaily,sescahbell:lit,'t!:eltd'ed: : Instead' ofanalyzing the strutl"tiutEfundsf: sifi9'ille waVe; adbi'ol'i;,. thestructure is su)j~ectEid foa' SE!t'i:e~s of waves', toaccount for the cyclic natuit's' of Bhe' loading.

A methodology, COnlplertientfng: uh'ef piie'\t:foU,s pUSh-over procedures, is ot1tlil1ed:'~dr. :trttsl]'tiAt:y aS~sessiifE!ntof structures under revised 'dtil's\])

This new approach to reassessment is being in-vestigated within a new project which is sponsored byseveral oil companies and engineering consultants.

straining of the material may occur. It does notrepresent global failure as such, but may prevent thestructure from reaching the shakedown state, or mayaccelerate the incremental collapse process (Figure5)

BASIC CONSIDERATIONS

Assume that the design conditions for a structurehave been changed so that the ULS criterion (firstmember failure) is no longer satisfied.

During the remaining life, the structure will beexposed to a large number of waves. As illustrated inFigure 1 the vast majority of the waves produce onlyelastic response. However, some few, large waves maycause elasto-plastic response; i.e. the maximum wavein the remaining life (design wave), and depending onthe load/strength "mismatch", the second highest wave,the third highest wave etc.

From wave statistics, it is shown (Gumbel, 1958)that in the order of 5 to 20 waves are likely toexceed 90% and 80% of the design wave height, respec-tively. Hence, the number of waves to be consideredthat cause plastic deformation, is small. In order toclarify the following discussion, the time history hasbeen ordered according to decreasing size in Figure 2.

For simplicity it may be assumed that the struc-ture is subjected to waves with the same magnitude asthe maximum wave/design wave. Further, it is shown inLuyties and Geyer (1987) and Hellan et al (1990) thatthe maximum load in the opposite direction duringpassage of the wave is smaller than the design waveload, in the order of 25-40%.

SHAKEDOWN THEORY

The following questions should be answered if astructure is subjected to variable loads that causeplasticity:

for a given structure and load domain Q, will thestructure reach shakedown avoiding incrementalcollapse and/or alternating plasticity for any loadpath contained within this domain?if the structure shakes down, will the resultingdeformations be of acceptable magnitudes?if the structure shakes down for a domain Q, whatis the safety agains incremental collapse or alter-nating plasticity (i.e. how much may the loaddomain be enlarged)?

The shakedown theorems state that, when ful-filled, a structure subjected to variable loads withunknown history but known load domain is safe withrespect to these failure situations. If the structureshakes down, further variable loading of the same (orsmaller) magnitudes does only cause elastic deforma-tions, i.e. the plastic deformations have becomestationary. Corresponding to this plastic field, astationary residual stress field p (x, t) = P (x,t sh d .) has also developed. Stationary fields may beformally expressed as (Stein et al, 1990)

When the structure is loaded up to the designload level, yielding takes place in the sections wherethe ULS criterion is violated. Redistribution offorces may also lead to yielding in other sections inthe structure. As shown in Figure 3, yielding causesa reduced overall stiffness, and produces permanentplastic deformations when the structure is unloaded.

lim P (x, t) - 0t - 00

lim p (x, t) - 0t - 00

(1 )

(2 )

These relations express whether shakedown occur.Also, the accumulated plastic work in the structuremust be bounded (Konig, 1987), or else incrementalcollapse/alternating plasticity occur.

Physically, the ability of materials and struc-tures to accumulate plastic work is limited due toexhaustion of ductility/low cycle fatigue, wheresignificant cracks have developed and sudden fracturesmay occur. Hence, the infinite term should be replacedby some finite, physically based, magnitude Wp'max.

If the structure has reached the shakedown statefor an arbitrary load path within the load domain, theyield criterion, Eq. (4), is satisfied everywhere inthe structure. This means that the external loads arecarried elastically, and the stresses in the structuremay be written as the sum of an elastic field and theresidual stress field, Eq. (6).

f[O(x, t))

(3 )

(4 )

(5 )

P dt dV < 00

o

00 (x)

f - 00

t=oof f 0V 0

F

When the structure is reloaded, yielding may takeplace once again. Within a specific load range, repe-titive actions of the load give smaller and smallerincrements in the plastic deformations. In the end,the structural behaviour may be fully elastic, butwith permanent plastic deformations in some parts ofthe structure, as illustrated in Figure 3. The perma-nent plastic deformations in the initial cycles inducea residual stress field in the structure. In thesubsequent cycles, this stress field counteracts theeffect of the wave load, and the structure reachesshakedown.

On the other hand, if the load/strength "mis-match" is higher, the plastic deformation incrementsmay not decrease during the load cycles. In this case,the accumulated displacements sooner or later grow solarge that the structure becomes unstable and failsdue to incremental collapse (Figure 4).

These two situations are also schematically illu-strated in Figure 6. The loading is represented by atotal load rosette or a load domain Q. There exists amaximum size of the domain corresponding to shakedownof the structure. For loads outside this domain, thestructure will fail due to incremental collapse.

So far, ductile material behaviour has beenassumed. However, fracture due to alternating plasti-city (low cycle fatigue) or extensive inelastic

o OE + P

f[OE (x, t) + PIx))

(6 )

(7 )

35

where a E denotes the el 1, suchthat

is satisfied fOJ; all loao.s defining Q, and forall x in thE) structure, then the stJ;ucture willshake down for the given 10ao. domain.

Note that p may be different from the exact p.

The load domain, Figure 6, may also be describedby the vertices (ve.ctors from the origin to thecorners of the hyper-parallellepiped Q). The followingtheorem derived by Konig and Kleiber (1978) may beapPlied in order to dis.creti~e the time variation:

B) Ifa given structure shakes down over a cyclicload path containing all. the vertices ~; of theload dom

Elasto-plastic element

differential equation for a beam subjected to axialforce and laterial bending loads at the beam ends

The stiffness matrices for the nonlinear elasticbeam element is derived as closed form solutions. Nonumerical integration over element length or overelement cross section is necessary.

Material nonlinearity is modelled by plastichinges introduced at member ends or at member midspan.The yield criterion is expressed in terms of stressresultants (axial force, shear force, torsion andbending moment).

EXAMPLE STUDIES

In Figure 8 some results are plotted. The vari-able load factor A is plotted on the ordinate axis,and the horisontal displacement of the upper beam isplotted along the abscissa-axis. In the first case anincremental collapse mechanism is rapidly developing.In the second case, where the maximum load level isreduced from 6.5 to 5, still an incremental collapsesituation is occurring. Finally, for a slight reduc-tion of load level, shakedown of the frame isobtained. Hence responds the structure elasticallyafter some elasto-plastic cycles for this loadpattern. The necessity of including nonlinearities inthe shakedown analysis was pointed out in (Skallerudand Larsen, 1989).

Two-storey Frame

In Figure 7 a two-story frame with horisontal andvertical loads varying between zero and a maximum isdepicted. The members have constant tubular cross-section. In addition to the variable loads Av and Ah'a pair of static axial loads N are present in thecolumns.

(11 )

( 12)

Al cos kx +Az sin kx + A3x + A4(compression)

Al e kx + Az e- kx + A3x + A4(tension)

v(x)

v(x)

N ,where k -E and N l.S positive in compression.

IEAl - A4 are generalized coordinates.

(13 ) Plane Jacket Structure

stiffness matrices for elements with plastichinges are modified according to plastic flow theory.

The consistency criterion states that the stateof forces lies on the interaction surface duringplastic deformation. This is expressed for an elastic- perfectly plastic hinge as

r = 0 defines a plastic state of forces while-1 ~ r < 0 reflects elastic cross sections. r = 0represents a surface in the stress resultant space.According to plasticity theory, this surface isconvex, which is a fundamental requirement for shake-down analyses (Skallerud, 1991).

The finite element model consists of 24 nodes and47 elements. Each member is modelled by one finiteelement. Initial deformations are not considered, andjoints are modelled as rigid. Local failure modes ofthe braces and joints are disregarded, but in-planeand out-of-plane deformations are considered. Thestructure is subjected to a topside load of 50 MN anda total horizontal load of 46.7 MN.

A plane model of a jacket with a height of 184.5m is shown in Figure 9 (Stewart et al, 1988). Typicalmember diameters are 0.9 - 1.0 m for the horizontalsand 1.3 m for the braces. Horizontals and braces ofthe upper panel are 0.6 m. Leg diameters range from1.3 m at the top to 6.0 m at mudline. D/t ratios varyfrom 26 to 57 for the horizontals, 42 to 66 for thediagonals and 41 to 121 for the legs.

(14 )(i=l, 2)

Two different cases are studied - each represent-ing a situation where the design conditions havechanged, thus requiring a reassessment of the struc-ture (Hellan, 1990).

where ~Si is the increment in forces and gil is thegradient of the yield surface at each node of theelement.

The elasto-plastic stiffness matrices for theelements can now be derived as

(15 )CaseCase

Increased environmental loadingDamaged/removed structural members

where ke is the elastic, tangential stiffnessmatrix.

Plastic hinges can be introduced at element endsand at midspan. Both gradual plastification and kine-matic hardening of the plastic hinges may be accountedfor in addition to local buckling, joint flexibility,and fracture.

In each case, a pushover analysis is run in orderto determine first member failure and ultimate capa-city of the structure. Subsequently, a cyclic load isintroduced, varying between 0 times the design load,and -0.3'0.

static pushover analysis gives a collapse load of1.79 times the design load. First yield occurs at 1.24times the environmental load, Figure 9b.

USFOS is based on an updated Lagrange formula-tion, with a simple Euler-Cauchy incrementation proce-dure and local element reference systems. Nodal co-ordinates are updated after each load increment, thusaccounting for large displacements and the "p-6"effect.

Increased environmental loading

The structure is analysed under cyclic loadingwith a loading factor 0 = 1.6.

Global deformations are shown in Figure lOcoAfter the initial loading, the increments in plasticdeformation are relatively constant. The accumulatedplastic displacement are slightly reduced (the incre-

37

menta'J.:pl.asticdisplacements hasopposite sign of thein it:ial ,u:t'ertime C lllusull1ptian:f:lll'X'co'mpl'e'X .'S'truC't:a:te.'s. A'm:a;j:orobjectiv:eofthet'esearc1').~p:ro}ect:i,st'h'er,e:f,oret:oes'bablish'somes:im,pl;j;fied 111u!\t'ht:i1,d ~,'t:o.:r'e'aAlo,e

certain range, shakedown takes place, i.e. the struc-tural response becomes elastic for a load level abovefirst yielding.

The cases indicate that offshore structures canbe utilized beyond the conventional ULS design limit(first member failure), and still fulfill the basicregulatory requirements (elastic behaviour) after anumber of cycles.

Although at a preliminary stage, the results sofar are very encouraging with respect to using acyclic approach for justifying acceptance of increasedutilization in connection with reassessment of struc-tures.

Application of the proposed methodology requireextensive investigations of cyclic, three dimensionalbehaviour of offshore structures, including possible

loa~ direction/load sequence effects. Further work isneeded to establish reliable failure criteria for com-ponent behaviour under cyclic loading in the elasto-plastic range, concerning local buckling, fracture andlow-cycle fatigue. These aspects will be addressed infurther within the new Reassessment project.

ACKNOWLEDGEMENT

The present study has been done as a part of theresearch project 'Reassessment of Marine structures',sponsored by the following companies : Shell, NorskHydro, Phillips, Saga, Amoco, Aker Engineering andOffshore Design. The authors wish to acknowlege theinput of ideas provided from the project sponsors. Inparticular, the contribution from project chairmanG.Stewart of Shell Research is highly appreciated.

REFERENCES

Borkowski, A. and Kleiber, M. (1980), "On a NumericalApproach to Shakedown Analysis of Structures", !&nUL..Meth. Appl Mech. Engng , Vol. 22, pp. 101-119.

ECCS (1978), "European Recommendations for Steel Con-struction".

Gumbel, E.J. (1958), "Statistics of Extremes", Colum-bia University Press, New York.

Hellan, 0., Amdahl, J., Farnes, K.-A. and Karunakaran,D.(1990), "Reassessment of Structures, FundamentalConsiderations", SINTEF Report STF71 A90023, Trond-heim.

Hellan, 0. (1990), "Cyclic Analyses of 2D JacketStructure", SINTEF Report STF71 A90017, Trondheim.

Konig, J .A. (1987), "Shakedown of Elastic-PlasticStructures", Elsevier Publishers.

Konig, J.A. and Kleiber, M. (1978), "On a new methodof shakedown analysis", Bull Acad. Pol Sci, Vol.26, pp. 165-171.

Lloyd, J.R. and Clawson, W.C. (1983), "Reserve andResidual Strength of Pile Founded Offshore Plat-forms", Proc. Int Symp Role Des Insp. RedundancyMarine Struct

39

Luyties, W.H. and Geyer, J.F. (1987), "The Developmentof Allowable Fatigue Stresses in API RP2A", Q.T.C.Paper 5555, Houston, USA.

Melan, E. (1938), "Der Spannungszustand eines Mises-Henchyschen Kontinuums bei veranderlicher Belast-ung", Akad Wiss Wien IIa, Vol. 147, pp. 73-78.

The Norwegian Petroleum Directorate (1984), Regula-tions for load-carrying structures for extractionor exploitation of petroleum", (in Norwegian),Stavanger.

Skallerud, B. (1986), "Nonlinear Effects on the Shake-down Load of Sway Frames and Continuous Beams", Div.Struct. Engng., NTH, Trondheim.

Skallerud, B. and Larsen, P.K. (1989), "NonlinearEff~cts on Shakedown of Sidesway Frames", J Struct~, ASCE, Vol. 115, pp. 221-227.

Skallerud, B. (1991), "Reassessment of Structures:Shakedown Theory and Relevance for Offshore SteelPlatforms", SINTEF Report STF71 F91005, Trondheim.

Stein, E., Zhang, G. and Mahnken, R. (1990), "Shake-down Analysis for Perfectly Plastic and KinematicHardening Materials", Stodola Session "Progress inComputational Analysis of Inelastic Structures.Udine, Italy.

Stewart, G., Efthymiou, M. and Vugts, J.H. (1988),"Ultimate Strength and Integrity Assessment of FixedFixed Offshore Platforms", Proc. Fifth BOSS Confer-~, Trondheim.

Scneide, T.H. et a1. (1986), "Collapse analysis ofFramed Offshore Structures", OTC Paper 5302,Houston, USA.

S0reide, T.H. et al. (1988), "USFOS Theory Manual",SINTEF Report STF71 F88036, rev. 1990-07-01, Trond-heiJU, Norway.

Waveheight

Waveheight

100 yeardesign Wave

Elastoplastic

Elasticresponse

}Elastoplastic

Figure 1 Wave load history Figure 2 Sorted wa;ve history

Elasticresponse

}E;lasto

.. plaStic

n

Shake down state (elastic)

Load

Design Load

Initial yield

/'/

//

Original designI Single waveRevised design conditions

Displacement

Load

Design Load

Initial yield

/'/

II

Original Oesign

J Single wave

Revised design conditions

Displacement

Load

Design Load

Initial yield

Figure 3 Shakedown behaviour

Low cycle fatigue fracture

Displacement

Figure 5 Low cycle fatigue

Figure 4 Incremental collapse

y rl9rth

........J.;;;;~~~ .,

,/ ~ shakedown "..... 1OO-year wind/waveload rosett~

,ii''''''''''''''''' ..... l.. ......4'"""'c:-+..-_..--1.f,-._.,..,..,...-~......:.;:----~-~e-a-st ..-O(X,y,t)

Figure 6 Load domain

40

l.:tVPO~~-----"';"'------i 2LM p constant

~~ -m~~J~__2L ~ O t=6..Croec tion: J D.263" }

.Figure 7

MEMBER 36 REMOVED

INTACT FRAMETOPSIDE i.O

MEMBERS 56 AND3[ REMOVED

0 .0 O.IS 0.20 I.~ 0.0\ 0.02 D.OJ 0.04 D.OS 0.111 0.01 I.M

4., 1M) 4,_ 1_).) Ial

Figure 8

~'2500 2500, ,

42

...J2.0III>~"0.31.5...J0...

CIIIe61.0l...J>Jj0.5

l9 400

-- --0.2 O.~ O.G 0.8 1.0

Ct.epLocement. 1m)..L

I 15 .. -I(b)

(a)Figure 9

41

0.,8. 0.8 1.0DYtP.LgOElmEl!'l~ (m'

'fOPS I Oecl Q.

o.!\l."".l,.,.....-=,.".,--=",....,"'"""'1--=,."...,,....,,,~"T'"--,..,F"'"--n-.,..,..,.0.0(l.E! 0.al.0Dl.E!pLcOE!IllE!A~ (m)

TOPSIDE I.B

0."0.2INTACT FRAME10 (p)

0."0.2

-,2.0 -,2.0 i\':!,QlD > > ]~ ~1l

11'00 g

.9 1.5 .;J .$1 ..:11 .'5-' -' l13 0-oJ ...c Ii!D ~E 6c 1.0 1.0 ~.O0 0S I. I.;J ;J> > >c ili cw W0.5 (l .5 0.5

0.8 o.a 1.0I;) l.E!P I"QC;:E!mE!!'l~ (m'

10 (a) : INTACT FR"Mg TOPS IDf;:1 .Q

P.6 .e.a l,ep\.iElP.\iaq.l!!lllen~ (Ill)

-'2;0 .

!"8.91.S.-'

i61.0S~

~ ... _-~---

Iii0.6 o.a 1.0p l.liIP I,.QC;:liIlll.!EI'n~ (m)

'"I

-,2.0~.~

]1.5

-,2.0 -'2.0 -,:a.ll !D III> ~ >!D -' -'

1l'8 1lD 0

0 1.5 .91 .5 0 1 ,$I-' ...I

...I ...I ...I0 i 0~ -tICII) II)E 61C 1 .0 61 .0 .0~ I. S;J ;J> > >ili ili I:w

0.10 0.5 0.5

Q.iiO.:a 0.8 a.a 1.0Ol.ElPLcC::li>mElnl (Ill')

11 (c) : MEMEl.ER$ :~9 AND~, REMOVED

0.6 C).a 1.0o"liIP LQO.II\E!A~ (m)

11 (b) : MEM8ER $i!3 REMOVED

0.6 0,8 1 .0O"",pI"QOl!!lllen~ (m)

TOPSIDE 1,0

0."0.211 (a): INTACT FR"ME

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42