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Ships and Offshore Structures

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Statistical assessment of seismic fragility curvesfor steel jacket platforms considering globaldynamic instability

M. Abyani, B. Asgarian & M. Zarrin

To cite this article: M. Abyani, B. Asgarian & M. Zarrin (2017): Statistical assessment of seismicfragility curves for steel jacket platforms considering global dynamic instability, Ships and OffshoreStructures, DOI: 10.1080/17445302.2017.1386078

To link to this article: http://dx.doi.org/10.1080/17445302.2017.1386078

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SHIPS AND OFFSHORE STRUCTURES, https://doi.org/./..

Statistical assessment of seismic fragility curves for steel jacket platforms consideringglobal dynamic instability

M. Abyani, B. Asgarian and M. Zarrin

Faculty of Civil Engineering, K. N. Toosi University of Technology, Tehran, Iran

ARTICLE HISTORYReceived July Accepted September

KEYWORDSIncremental dynamicanalysis; seismic; fragility;offshore; jacket platform

ABSTRACTMany probabilistic studies in the field of offshore structures assume that the drift demand distributes log-normally around its median at all the intensity levels. However, this assumption may not be formally val-idated, and the purpose of this study is to investigate the assumption of lognormal distribution of thedrift demand for the fixed offshore platforms, using Anderson–Darling goodness of fit test. The lognor-mal hypothesis for the drift demand can be investigated at two different regions as: (1) low intensity levelswithout any collapse case and (2) higher intensity levels with collapse cases. To this end, both the samplemedian and sample geometric mean are considered as the estimators of lognormal central tendency. Theresults indicate that the lognormal hypothesise is accepted based on the sample geometric mean at allthe intensity levels. Nevertheless, the lognormal hypothesise is rejected at some intensity levels if the sam-ple median is the central estimator.

1. IntroductionEarthquake is one of the most destructive natural disasterswhichmaymake drastic damages to the existing structures. Dif-ferent randomnature of earthquake such as occurrence time andlocation or seismic wave propagationmakes it quite complicatedfor the engineers to anticipate the exact seismic behaviour ofthe structures. However, after the 1994 Northridge and the 1995Kobe earthquakes, significant progress was made in earthquakeengineering by Federal Management Agency (FEMA) and SAC(joint venture of Structural Engineers Association of Califor-nia (SEA), Applied Technology Council (ATC), Consortium ofUniversities for Research in Earthquake Engineering (CUREE))projects. This highly efficient project has been proposed inFEMA-350/351 guidelines (SAC/ FEMA-2000a, 2000b).

The analytical framework of seismic reliability evaluationhas been widely expanded by Jalayer and Cornell (Cornellet al. 2002; Jalayer 2003). They derived closed form expressionsfor the probability of exceeding a limit state considering bothaleatoric and epistemic uncertainties in structural and seismicAnalyses. This framework becamemore simplified in aDemandand Capacity Factored Design (DCFD) format (Cornell et al.2002; Jalayer 2003) which is quite similar to the familiar Loadand Resistance Factor Design (LRFD) format (AISC 2003). Thisformatmakes it possible to calculate the seismic reliability of thestructure at each selected confidence level.

In this process, one of the most fundamental assumptionsintroduced by Shome (1999) is that the maximum interstorydrift ratio (MIDR) distributes lognormally at each level of spec-tral acceleration. It could be easily proven that for a perfectly log-normal random variable (a lognormal population with infinitesample size), the mean of the logarithm of that variable is equalto the logarithm of the median of the same variable (Benjaminand Cornell 1970; Soong 2004). As a result, MIDR demand

CONTACT M. Zarrin mo_zarrin@yahoo.com

distributes lognormally around its median at each level of spec-tral acceleration (Sa) (Shome 1999). However, in a lognormalsample with a finite sample size, the logarithm of the samplemedian is not exactly equal to the mean of the logarithm ofthe same sample, and consequently, the mentioned assumptionmight have some approximations. On the contrary, for any arbi-trary randomvariable the sample geometricmean exactly equalsthe mean of the logarithm of that sample (Shih and Binkowitz1967). In this respect, Abyani et al. (2017) compared the analyt-ical framework of seismic reliability evaluation of steel momentframes based on the sample median and the sample geometricmean as the index of central tendency. The results of their studyillustrated that the sample geometric mean could lead to moreaccurate results.

From another perspective, in the last two decades, a lot ofeffort has been made to evaluate and improve the performance-based assessments of the jacket type offshore platforms (JTOPs)(Hasan et al. 2010; Jahanmard et al. 2015; Elsayed et al. 2016).In 1996, Det Norske Veritas (DNV 1996) published a guide-line report for the offshore structural reliabilitywhich comprisedexperience and knowledge on the application of probabilisticmethods to structural design and provided advice on prob-abilistic modelling and structural reliability analysis of jacketstructures. In another study by Jahanmard et al. (2015), waveendurance time (WET) was addressed as an applicable methodfor performance-based evaluation of fixed offshore platformsunder extreme waves. In this research, artificial wave recordscalled wave functions were designed so that their excitationsgradually increase with time. Consequently, the main advan-tage of this approach was that it could assess the structuralperformance under various wave load conditions through asingle time-history analysis. Elsayed et al. (2016) presented anew method for reliability assessment of a fixed offshore jacket

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platform against earthquake collapse. They computed the prob-ability of platform collapse under seismic loading using a finiteelement reliability code. The first and second order reliabilitymethods were used to calculate the safety indices, which couldbe compared with the target safety levels in offshore platformdesign codes.

Additionally, uncertainty modelling with the nonlineardynamic analysis of JTOPs was discussed for how to account forthe different uncertainties in the reliability assessments. Sincejacket platforms may have inelastic behaviour during strongground motions, it is necessary to use advanced structuralanalysis methods such as incremental dynamic analysis (IDA)(Vamvatsikos and Cornell 2002). Asgarian and Ajamy (2010)studied the seismic performance of the JTOPs, employing IDAfor the structural analysis. They used the story drift as the engi-neering demandparameter (EDP) andfirstmode-spectral accel-eration as the intensity measure (IM). Golafshani et al. (2011)proposed the method of probabilistic incremental wave anal-ysis (PIWA) to evaluate the performance of JTOPs subject-ing to sever wave loadings. In this approach, both static anddynamic wave analyses were implemented to estimate the dis-tribution of wave height intensities. Also, an efficient combina-tion of Latin hypercube sampling (LHS) (McKay et al. 1979) andsimulated annealing (SA) technique (Vorechovsky and Novak2009) was employed to reduce the amount of computationalexpenses. Further, Ajamy et al. (2014) introduced a comprehen-sive interaction IDA method to incorporate different sources ofuncertainties associated with seismic load, modelling param-eters and soil properties in the stochastic seismic analysis ofJTOPs. In order to propagate these uncertainties, they used thesame combination of LHS and SA technique tomodel the corre-lation of the uncertain parameters such as yield strength, elastic-ity modulus, shear wave velocity, shear modulus reduction anddamping ratio. In another study, El-Din and Kim (2014) devel-oped a simple methodology for seismic life cycle cost (LCC)estimation of steel jacket platforms. They utilised equivalentsingle degree of freedom system instead of the main struc-ture, and eliminated the full IDA and fragility analysis. Instead,approximate fragility curves and localised IDA curves were usedas well as a probabilistic simple closed-form solution for lossestimation.

In all these studies (Asgarian and Ajamy 2010; Golafshaniet al. 2011; Ajamy et al. 2014; El-Din and Kim 2014), it wasassumed that the structural demand conditional on a seismicintensity or a wave height level follows a lognormal distribu-tion around its sample median. However, this paper aims toinvestigate the validity of lognormal hypothesis for the struc-tural demand of JTOP. In this regard, Anderson–Darling (AD)goodness of fit test (Anderson and Darling 1954) has beenused to check whether the lognormal distribution is suitablefor the structural demand of fixed offshore platforms or not.Furthermore, it is intended to compare the accuracy of seis-mic fragility curves based on both the sample median andthe sample geometric mean as the statistical index of lognor-mal central tendency, in two regions: (1) where no records hasreached the global dynamic instability and (2) at higher inten-sity levels where the records consecutively reach their collapsecapacities.

2. Preliminary probabilistic considerationsThis section aims to discuss some preliminary probabilistic sub-jects required for the evaluations of this study.

2.1. Central tendency in lognormal distributionTheprobability density function of a lognormal randomvariable(f) is defined as (Soong 2004):

f (D;μ, σ ) = 1Dσ

√2π

exp(

− (Ln(D) − μ)2

2σ 2

)(1)

where D is a lognormal random variable with the statisticalproperties equal to μ (the mean of the natural logarithm of therandom variableD) and σ (the standard deviation of the naturallogarithm of the random variable D).

It can be proved that for a perfectly lognormal random vari-able such as D (a lognormal population with infinite samplesize), the natural logarithm of the median of D is equal to μ

(Benjamin and Cornell 1970; Soong 2004). This is the reasonwhy the sample median has been considered as the lognormalcentral tendency in many previous studies (Cornell et al. 2002;Jalayer 2003; Asgarian and Ajamy 2010; Golafshani et al. 2011;Vamvatsikos 2013; Ajamy et al. 2014; El-Din and Kim 2014).Nevertheless, the natural logarithm of the sample median of alognormal random variable with a finite sample size is not equalto themean of the natural logarithmof that sample. On the otherhand, the natural logarithm of the sample geometric mean ofany arbitrary random variable is always equal to the mean of thenatural logarithm of the same sample, regardless of its samplesize (Abyani et al. 2017). Hence, regarding the maximum likeli-hood estimation (MLE) method, the sample geometric mean isregarded as a more efficient estimator of lognormal central ten-dency (Walpole et al. 2002). In other words, for the lognormalvariableD, the sample geometricmean is theMLE estimator andthe sample median is the rank estimator of the exponential ofμ(Shih and Binkowitz 1967).

2.2. Seismic fragility curvesSeismic fragility evaluation is the core of probabilistic safetyassessments of structures, which can be expressed through afragility curve. A fragility function demonstrates the conditionalprobability that the structural capacity fails to resist the struc-tural demand, given the seismic intensity (Celik and Ellingwood2010). Considering the IDA curves, the MIDR demand fragilitycan be assessed at each level of spectral acceleration Sa as theparameter of IM. In case the global dynamic instability is nothappening at the given IM level, the analytical fragility func-tion is calculated by the two parameter lognormal cumulativedistribution function (CDF) (Shome 1999; Cornell et al. 2002;Jalayer 2003). Nonetheless, at higher IM levels where the groundmotion records reach their collapse capacities, the three param-eter distribution proposed by Shome (1999) is capable of esti-mating the distribution of the drift demand with cases of globaldynamic instability. In fact, the analytical expression of the threeparameter distribution has been derived based on the disaggre-gation of the demand response values into two groups: non-collapse (NC) and collapse groups. The mean of the natural

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logarithm of the NC drift demands d|NC, Sa at the given Sa level(μ(Ln(d|NC, Sa))), the standard deviation of the natural loga-rithm of the NC drift demands at the given Sa level (βd|NC,Sa =σ (Ln(d|NC, Sa))) and the probability of NC response valuesPNC|Sa at the given Sa level are the three parameters of this distri-bution function. It is noted that the probability ofNCdrift valuesat the given Sa level equals the ratio of the number of NC recordsto the number of all the records at the same Sa level (Shome 1999;Jalayer 2003), which is always less than or equal to unity.

F(d|Sa) = �

(ln(d) − μ(Ln(d|NC, Sa))

βd|NC,Sa

).PNC|Sa (2)

In this equation, � represents the standard normal CDF.

2.3. Anderson–Darling goodness of fit testAD test can be regarded as a modification of the Cramer-VonMises test, and is one of the most powerful goodness of fit tests(Anderson and Darling 1954) based on empirical distributionfunction (EDF). In fact, AD test was developed as an alternativeto other statistical tests for detecting sample distributions’ depar-ture from normality (Engmann and Cousineau 2011). The ADtest statistic (STAD) is categorised as the quadratic class of theEDF statistic which is based on the squared difference betweenthe empirical CDF (ECDF) and the hypothesised analytical dis-tribution F as Equation (3). For sample values sorted in ascend-ing order of magnitude, denoted by x1, x2, ..., xn the ECDF of xiis calculated by i

n , where i is the rank of x.

STAD = n∫ +∞

−∞[ECDF(x) − F(x)]2.ψ(F(x)).dF(x) (3)

STAD is the average of the squared discrepancy[ECDF(x) − F(x)]2 weighted by ψ(F(x)), and ψ indicatesa nonnegative weight function that could be derived from dif-ferent expressions such asψ = [F(x)(1 − F(x)]−1. The AD testis more sensitive to deviations in the tails of the distributions(Anderson and Darling 1954), and is more powerful than theKolmogorov–Smirnov (KS) test (Benjamin and Cornell 1970;Razali and Wah 2011).

3. Description of the case study platformThe considered sample platform is one of the two newlydesigned similar platforms located in the PersianGulf. This plat-form was planned and designed in 2010 and was installed in2012 in water depth of about 75 m. The platform comprises afour-leg jacket supporting a wellhead deck, which was designedand analysed in accordance with the requirements of the Amer-ican Petroleum Institute (API RP2A-WSD 2007).

The steel jacket is composed of six horizontal plan frames thatform an unbraced story and four braced stories, two vertical legsand two legs battered to 1:10 in direction y. The jacket is approx-imately 84 m high from mud line up to top of the jacket. Theutilised lateral resisting systems of the jacket are the V-bracingand inverted V-bracing systems, which constitute a MultistoryX-bracing system at the second and fourth horizontal plan fram-ings (See Figure 1). The Jacket is fixed to the ground by fourthrough leg piles. The gap between pile and leg is filled withgrout to provide a composite section termed grouted section.

Figure . Perspective plot of the HE platform (a) D sacs model, (b) D openSEESmodel and (c) Row view. (This figure is available in colour online.)

All top side and jacket masses including structural andhydrodynamic masses have been considered as concentratedmasses at the element joints. Accordingly, all top side and jacketloads are applied at joints as equivalent point loads.

During the design phase, a three-dimensional space framecomputer model of the platform with all primary and sec-ondary componentswere created by using the structural analysiscomputer system (SACS) suite (Figure 1(a)). Herein, a numer-ical model of the aforementioned sample platform has beendeveloped using OpenSEES software (Mazzoni et al. 2007). Theschematic configuration of the model created in OpenSEES isalso demonstrated in Figure 2(b,c).

The soil profile was determined based on the results of acomprehensive geotechnical investigation including some bore-holes and a piezocone penetration testswithin the location of theproject. The soil stratigraphy conditions disclosed by the bore-hole performed at the platform location are medium dense siltyor clayey sand to around 2.0 m depth frommudline, followed byfirm locally sandy clay to around 3.8m. Firm becoming stiff clay,generally of intermediate to high plasticity was encountered toapproximately 70 m followed by very stiff becoming hard clay ofhigh plasticity to the completion depth of each borehole.

4. Numerical modelling proceduresIn this paper, the nonlinear beam-column element (distributedplasticity) has been used to model the jacket and pile elements,which is based on the force formulation (Spacone et al. 1996).In this model, the element is subdivided longitudinally intoa number of segments, and the fibre discretisation approachhas been employed to numerically model the cross section ofeach segment. The considered stress–strain relationship for thesteel material of the pile and jacket members is based on theMenegotto-Pinto model (Menegotto and Pinto 1973). One of

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4 M. ABYANI ET AL.

Figure . Site-specific seismic hazard curve for the fixed jacket platform located in Persian Gulf. (This figure is available in colour online.)

the merits of this type of material is the gradual transitionfrom linear to nonlinear range of response. Moreover, the cyclicbehaviour of this material is fairly consistent with experimentaltests including Baushinger effects.

The nonlinear beam-column element used in this researchis limited to small deformations in the basic system of the ele-ment. The large displacements are taken into consideration byembedding the basic system of the element in a corotationalframework, as proposed by De Souza (Souza 2000). Also, thenonlinear geometry effects can be readily incorporated duringthe transformation of the basic response quantities to the globalreference system of the structural model.

In order to include global buckling in the braces of JTOP,the brace is modelled with two force-based elements and an ini-tial camber equal to 1/1000 of brace length on the mid node ofthe brace. Considering this technique along with co-rotationalformulation for coordinate transformation, buckling and post-buckling behaviour of braces can be taken into consideration,accurately.

Asgarian et al. (2005) modelled the post buckling and alsohysteretic behaviour of pinned and fixed tubular struts whichwere subjected to cyclic loading, using the force-based fibrebeam column element. Asgarian et al. (2006) andHonarvar et al.(2008) used the same element in their studies to predict the seis-mic response of tested X-braced jackets. Furthermore, Alanjariet al. (2009) and Uriz et al. (2008) worked on the seismic per-formance of individual tubular strut tested by Black et al. (1980)upon buckling in compression as well as their capability in dis-sipating energy in consecutive cycles utilising the same elementas the case here. All the aforementioned studies confirmed theefficiency of nonlinear beam column element for the steel jacketplatform.

A beam on nonlinear Winkler foundation (BNWF) numer-ical model is created to perform uncoupled seismic soil-pile-superstructure interaction (SPSI) analysis. In order tomodel the pile elements in OpenSEES, nonlinear beam-column element is again employed. Zero length elements whose

force-deformation constitutive behaviour representing soil nearfield springs are connected to every nodes of pile below thesoil surface. Uniaxial p-y and t-z and q-z material objects inthe lateral and vertical directions were assigned to these zerolength elements. The PySimple1 uniaxialmaterialmodel class inOpenSEES has been used tomimic the lateral behaviour of near-field soil. This element is formulated and verified with experi-mental data by Boulanger et al. (1999). The t-z materials wereincorporated into zero length elements to apply skin friction onthe pile, and a q-z element is also assigned under the pile tomodel the end bearing reaction.

Soil profile horizontal response should be derived as a func-tion of depth to vertically propagating shear waves, as thefirst step of uncoupled SSPSI analysis. To model the saturatedsoil deposit dynamic response, a two-phase material based onBiot theory for porous media has been applied. In OpenSEES,four-node plane strain quadUp element implements a simpli-fied numerical formulation of this theory known as U-P for-mulation. In addition, in order to simulate the stress–strainbehaviour of the sand and bay mud, the pressure-dependentmulti yield (Yang et al. 2003; Mazzoni et al. 2007) materialand the pressure-independent multi yield (PIMY)material havebeen used, respectively. The procedure of conducting free fieldsite response analysis and the soil-pile-structure interactionmodelling approach implemented in this study has been veri-fied with experimental data in a previous work of the authors(Asgarian et al. 2013).

Damping as presented by Equation (4) is referred to Rayleighproportional damping (Charney 2008), which has been fre-quently used in previous studies (Asgarian and Ajamy 2010;Ajamy et al. 2014).

C = a0.M + a1.K (4)

The first term of the aforementioned equation is the massproportional term M and the second one is the stiffness pro-portional term K. Rayleigh damping is supposed to simulate the

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SHIPS AND OFFSHORE STRUCTURES 5

Table . Seismic characteristics of the ground motion records adopted for IDA (Baker et al. ).

REC No NGA REC No Earthquake name Year Station M R (km) Vs (m/s)

San Fernando Lake Hughes # . . Loma Prieta Gilroy array # . . Kocaeli, Turkey Izmit . . Northridge- LA – Wonderland Ave . . Imperial Valley– Cerro Prieto . . Hector Mine Hector . . San Fernando Pasadena – Old Seismo Lab . . Duzce, Turkey Lamont . . Hector Mine Heart Bar state Park . . Chi-Chi, Taiwan TCU . . Chi-Chi, Taiwan- TCU . . Coyote Lake Gilroy array # . . Taiwan SMART() SMART E . - Irpinia, Italy- Bagnoli Irpinio . . Loma Prieta San Jose – Santa Teresa Hills . . Irpinia, Italy- Bisaccia . . Chi-Chi, Taiwan TCU . . Kocaeli, Turkey Gebze . . Northridge- Pacomia Dam (downstr) . . Denali, Alaska Carlo (temp) . . Helena, Montana- Carroll college . - Northridge– Vasquez Rocks Park . . Chi-Chi, Taiwan WNT . . Loma Prieta Golden Gate Bridge . . Loma Prieta UCSC . . Victoria, Mexico Cerro Prieto . . Northridge– Santa Susana Ground . . Loma Prieta Gilory – Gavilan Coll . . Duzce, Turkey Mudurnu . . Northridge– Burbank – Howard Rd. . . Chi-Chi, Taiwan- TCU . . Chi-Chi, Taiwan- TCU . . Loma Prieta UCSC Lick Observation . . Loma Prieta Gilory Array # . . Northridge– LA Dam . . Northridge– LA . . Sitka, Alaska Sitka Observatory . . Northridge– LA – Chalon Rd . . Loma Prieta Belmont – Envirotech . . Chi-Chi, Taiwan TCU . .

condition that the structure is submerged in a viscous fluidwhenthe fluid imposes drag force on the structure .This is the rea-son of using Rayleigh damping in the numerical model of theoffshore structures. a0 and a1 are the proportional coefficientswhich are determined by specifying the damping ratio in anytwo vibration modes (Charney 2008). It should be emphasisedthat when the system responds inelastically, the stiffness termindicated in Equation (4) changes. Hence, in order to deal withthe inelastic response, the proportional coefficients a0 and a1are computed on the basis of the initial stiffness, and the damp-ing matrix is updated each time the tangent stiffness changes. Inthis case, the damping matrix will be classical at each step in theanalysis as the currentmode shapes will diagonalise the updatedstiffness matrix (Charney 2008).

5. Record selection and site specific seismic hazardcurveIn this investigation, a suite of 40 unscaled three-componentfar field ground motion records are selected so that their hor-izontal response spectra fairly match the target mean and stan-dard deviations in full logarithmic coordinates predicted for amagnitude of 7 strike-slip earthquake at a distance of 10 km(Baker et al. 2011). The target response spectrum represents

the high-seismicity sites that may experience strong groundmotions from mid to large-magnitude earthquakes at close dis-tances. The groundmotions were selected tomatch this target atperiods between 0 and 5 sec. The site Vs30 (average shear wavevelocity in the top 30m)was assumed to be 750m/s representinga rock site, which is supposed to be used as bedrock level groundmotions for site response analysis (Baker et al. 2011). Since theground motion set is neither structure-specific nor site-specific,the selected records have a variety of spectral shapes over awide range of periods, and are capable ofmodelling the aleatoricuncertainties in the structural analyses. The seismic character-istics of the 40 selected groundmotions are tabulated in Table 1.

Further, the site-specific probabilistic seismic hazard curvethat corresponds to the fundamental period of the structure(T1 = 2.15 s) has been calculated by probabilistic seismic haz-ard analysis (PSHA) (Bazzurro andCornell 1999), and plotted inFigure 2. The corresponding Sa for each value of mean annualfrequency of exceedance can be obtained from this figure.

6. Results and discussionsAs the first result presentation, the IDA curves as well asthe sample median and the sample geometric mean of theNC demand responses and the points associated with global

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Figure . (a) IDA curves, sample median and sample geometric mean of non-collapse records and the GI limit state and (b) summarised IDA curves (%, %, %, %,%, % and % percentiles). (This figure is available in colour online.)

instability (GI) limit state (where the IDA curves flatten) havebeen all illustrated in Figure 3(a). In addition to this, the sum-marised IDA curves (10%, 16%, 25%, 50%, 75%, 84% and 90%percentiles) are depicted in Figure 3(b). Since, in this study thefailure mode of the numerical model – in most of the nonlinearanalyses – is the portal frame mechanism in piles, the pile drift

per unit length of the pile has been considered as the EDP.More-over, the 5% damped elastic spectral acceleration at the funda-mental period of the structure (Sa(T1,5%)) has been chosen asthe IM.

As is seen, the sample median and the sample geometricmean of the NC demand responses are significantly different,

Figure . (a) AD P-values based on the sample median and sample geometric mean and (b) ISE values based on the sample median and sample geometric mean. (Thisfigure is available in colour online.)

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SHIPS AND OFFSHORE STRUCTURES 7

Figure . Empirical and analytical seismic fragility curves of drift demand at (a) Sa = 0.12g, (b) Sa = 0.28g, (c) Sa = 0.54g, (d) Sa = 1.14g, (e) Sa = 2.62gand (f ) empiricaland analytical seismic fragility curves of the IM of GI limit state. (This figure is available in colour online.)

especially at the IM levels where several records have reachedtheir GI limit state. Therefore, considering either of them as theestimator of lognormal central tendency could point to differentresults in seismic fragility assessment of the sample platform.

Prior to obtaining the seismic fragility curves, it is intendedto investigate the validity of the lognormal hypothesise for thedrift demand at each IM level. To this purpose, AD goodness offit test (Anderson and Darling 1954) has been employed. In fact,the lognormal hypothesise for the drift demand at each IM levelis rejected if the AD P-value is less than the probability of Type1 error which is referred to as the significance level α(Andersonand Darling 1954; Soong 2004). Commonly α is equal to 0.01,0.05 or 0.10 (Soong 2004), and in this study it is assumed thatα = 0.05, 0.10. Regarding both the samplemedian and the sam-ple geometric mean as the lognormal central tendency, the ADP-values of the drift demand at each IM level are illustrated inFigure 4(a).

It is evident from the figure that the AD P-values basedon the sample geometric mean are greater than the AD P-values based on the sample median at all the IM levels, whichindicates that the lognormal hypothesise is rejected athigher significance levels if the sample geometric mean isconsidered as the statistical estimator of lognormal central

tendency. Moreover, it is observed that for α = 0.10, the log-normal assumption is not accepted at several IM levels wherethe AD P-values are less than 0.10 (the black dashed line) ifthe sample median is supposed to be the estimator of centraltendency. Even in case α = 0.05 (the green dashed line), thereare still a few IM levels around Sa = 2.2g where the lognormalassumption is rejected based on the sample median.

In the next step, the analytical seismic fragility curves ofdrift demand based on both the sample median and the sam-ple geometric mean as well as the empirical fragility curves havebeen assessed for two IM levels (0.1230 and 0.2751 g) corre-sponding to the 400 and 2500 year return period earthquakesas recommended by API RP2A-WSD (2007). In addition, theseismic fragility of drift demand have been computed for theIM levels at which 16%, 50% and 84% ground motion recordsreach their global dynamic instability limit states. These IMlevels (0.54, 1.14 and 2.62) correspond to the 17,783, 421,696and 35,774,322 year return period earthquakes, respectively (seeFigure 2). Also, the analytical and empirical fragilities of col-lapse limit state (the IM values associated with onset of GIlimit state) have been calculated for the sample platform. Allthe aforementioned seismic fragility curves are demonstrated inFigure 5.

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It is emphasised that the lognormal assumption is acceptedfor all the IM levels at which the analytical seismic fragilitycurves are presented in Figure 5. As is observed, the fragilitycurves do not reach unity at the IM levels equal to 0.54, 1.14 and2.62 g as several ground motion records have reached their col-lapse capacities at these IM levels. As described in Section 2.2,PNC|Sa is equal to unity if no record has reached the GI limitstate. Otherwise, the term PNC|Sa is definitely less than unity andso does the maximum value of the fragility function. Addition-ally, it can be seen that the analytical fragility functions based onthe sample geometric are closer to the empirical fragility func-tions. In order to verify such observation with a quantitative cri-terion, the integrated squared error (ISE) – known as L2 distance– (Clarke et al. 2012) between the lognormal CDF and the ECDFof the drift demand based on the sample median and the sam-ple geometric mean (at all the IM levels) have been assessed byEquations (5) and (6) using the compositeGauss–Legendre inte-gration technique (Hildebrand 1956), and shown in Figure 4(b).

ISE[η] =∫ +∞

−∞

[ECDF(x) − CDF[η](x)

]2.dx (5)

ISE[GM] =∫ +∞

−∞

[ECDF(x) − CDF[GM](x)

]2.dx (6)

The indices [η] and [GM] following each parameter refer tocalculation of that parameter based on the sample median andthe sample geometric mean, and the variable x is referred to dif-ferent levels of the MIDR demand.

Figure 4(b) reveals that the ISE values based on the samplegeometric mean are less than the ISE values based on the sam-ple median at most of the IM levels, which could be consid-ered as other evidence for the sample geometric mean or againstthe sample median. Analogously, the ISE value of the IM of GIlimit state based on the sample geometric mean is less than theone based on the sample median (these values are illustrated onFigure 5(f)).

7. ConclusionsIn this paper, the validity of lognormal distribution of the driftdemand and spectral acceleration capacity for the JTOPs hasbeen investigated by the powerful AD goodness of fit test. In thisregard, both the samplemedian and the sample geometric meanhave been considered as the estimator of lognormal central ten-dency. All the conclusions are drawn as follows:

� The lognormal hypothesise of the spectral accelerationsassociatedwith onset of collapse and the drift demand at allthe IM levels for the sample offshore platform is acceptedby the AD goodness of fit test if the sample geometricmean is considered as the lognormal central tendency. Onthe contrary, lognormal assumption based on the samplemedian as the index of central tendency can be rejectedfor the sample jacket platform at some IM levels.

� The discrepancies between the sample median and thesample geometric mean of the NC drift demands grow asthe IM value increases. The reason is that the records reachtheir GI limit states at higher IM levels.

� The AD P-values of the drift demand of the sample off-shore platform based on the sample geometric mean aregreater than the AD P-values based on the sample medianat all the IM levels. This could be considered as substantialevidence for the sample geometric mean and against thesample median.

� The AD P-value of the spectral accelerations associatedwith the onset of collapse based on the sample geometricmean is again greater than the one based on the samplemedian.

� The ISE values between the empirical and analyticalfragility of the drift demand of the jacket platform basedon the sample geometric mean are less than the ones basedon the sample median at most of the IM levels.

Disclosure statementNo potential conflict of interest was reported by the authors.

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