Engineering Structures 92 (2015) 172–185

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Approximation of the maximum dynamic stress range in steel catenaryrisers using artificial neural networks

http://dx.doi.org/10.1016/j.engstruct.2015.02.0250141-0296/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Fax: +61 8 6488 1044.E-mail address: lucile.queau@gmail.com (L.M. Quéau).

Lucile M. Quéau ⇑, Mehrdad Kimiaei, Mark F. RandolphCentre for Offshore Foundation Systems, The University of Western Australia, Crawley, WA 6009, Australia

a r t i c l e i n f o

Article history:Received 9 February 2015Revised 11 February 2015Accepted 12 February 2015Available online 25 March 2015

Keywords:Steel catenary risersFatigueSensitivity analysesDesign of experimentArtificial neural network

a b s t r a c t

Adequate assessment of fatigue damage in steel catenary risers (SCRs) is essential, and usually evaluatedwith time consuming numerical analyses. Simplified design strategies would improve the efficiency ofthe screening tasks in the early design stages.

As part of on-going research aiming to define a simplified fatigue design procedure for SCRs in thetouchdown zone (TDZ), the sensitivity of fatigue damage to various parameters is explored using a largedatabase (>40,000 cases). An approximation of the maximum stress range in the TDZ is established usingseveral artificial neural networks and predicts well the fatigue life of selected example SCRs.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction focusing on defining an approximation of the maximum dynamic

The depletion of shallow water hydrocarbon reserves hasfocused the oil and gas industry increasingly on reserves in deepwater, where steel catenary risers (SCRs) represent one of the mostwidely used option to convey the oil and gas from subsea to thesurface. The main drawbacks of SCRs however is their highsensitivity to environmental loading that generates fatigue dam-age, especially in the area where the riser is in contact with theseabed, namely the touchdown zone (TDZ) (e.g. [1,2]). An accurateestimation of SCR fatigue life is fundamental to ensuring riserintegrity over the life of the project while keeping costs low.

The fatigue damage is often estimated through time domainanalyses to account for SCR nonlinearities (material and geo-metrical for instance) and by performing a series of time consum-ing numerical simulations [3–5]. A riser design standard [5]therefore encourages the use of simplifying techniques, especiallyfor the early stages of design, to improve the efficiency of computeranalyses and support engineering judgement. It states that‘‘numerous simplified analyses will normally produce more informa-tion regarding overall static and dynamic system behaviour whencompared to a reduced number of advanced analyses.’’

In light of these facts and recommendations, the authors havebeen aiming to develop a simplified riser fatigue analysis procedurefor the early stages of SCR design, avoiding the need to perform timeconsuming analyses. The present paper details part of that research,

stress range of SCRs in the TDZ (Max DrTDZ) valid for a wide rangeof input parameters. Max DrTDZ is used as, together with the num-ber of cycles of each magnitude of motion applied, it controls thefatigue damage. The method followed to develop the approx-imation is the same as in Quéau [6] and Quéau et al. [7] where anapproximation, namely ‘9-ANNs static approximation’, was definedfor the maximum static stress range in the TDZ.

The sensitivity of Max DrTDZ to the variation of design inputparameters was investigated by performing a large amount ofnumerical analyses. A similar in-house automation subroutine aspresented by Quéau et al. [7] was used for the pre and post pro-cessing tasks. It consists in linking the marine analysis softwareOrcaFlex [8] with the optimisation software modeFRONTIER [9]to generate a large database of SCR cases selected through designof experiment (DoE) techniques. A case is defined as a SCR config-uration under a given dynamic displacement. The flowchart of theautomation subroutine is shown in Fig. 1 using the notation fromQuéau et al. [10] as adopted hereafter. The dimensionless groupsshown to influence SCR stress range in previous work [10] consti-tute the input parameters so that the function f to be modelled isdefined as

MaxDrTDZ

E¼ f

HDz

;Dhm;HT

ffiffiffiffiffiffiffiffiffiffiqsteel

E

r;Do

Dz;Do

wt;

pEDz

; m;To

EDz2 ;l;ks

E;

�

CD;CA;qsteel

qwater;gDzqsteel

E; b

�ð1Þ

With

L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185 173

qsteel, qwater

steel and water densities Do, wt riser outer diameter and wall thickness E Young’s modulus g gravity acceleration H, T heave amplitude and period of the input

motion

ks soil stiffness p unit submerged weight To horizontal tension component Dz vertical difference between hang-off

point and seabed

s arc length (measured from hang-off

point)

H/Dz = p2 dimensionless riser displacement

amplitude

Dhm = p3 angle of the motion relative to the hang-

off angle (hHO)

HT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqsteel=E

p= p4

dimensionless riser displacement

velocity

Do/Dz = p5 dimensionless riser outside diameter Do/wt = p6 riser outside diameter to wall thickness

ratio

p/(E Dz) = p7 dimensionless riser unit submerged

weight

m = p8 Poisson’s ratio To/(E Dz2) = p9 dimensionless riser tension l = p10 soil friction coefficient ks/E = p11 dimensionless soil stiffness CD = p12, CA = p13 drag and added mass coefficient qsteel/qwater = p14 relative steel and water densities g Dz qsteel/E = p15 dimensionless water depth s/Dz = p16 location along the SCR b = p17 angular position on the SCR

circumference

An illustration of an example SCR configuration and some of theindividual input parameters are shown in Fig. 2.

The response surface method is applied with the artificial neu-ral network (ANN) to find the function approximating the relation-ships between the various design input parameters and the output.The aim is to define a function that could approximate Max DrTDZ

results from OrcaFlex software within ±5% relative error, which isregarded as negligible error for practical applications [11]. Theuse of ANN is not common for SCR design, although the ANNapproach has been applied successfully in other engineering fields,e.g. in geotechnical engineering [12], mechanical engineering [13]or in civil engineering [14,15].

The same simplifying assumptions and parameter definitions asin Quéau et al. [7] are used here. The study is limited to the SCRresponse under in-plane motions only with the loading appliedby imposing a sinusoidal displacement to the floating vessel. Thecurrent profile in the sea column, the rotational stiffness at thehang-off point, the flow rate of the contents, the coating and thestructural damping are not taken into account. A linear soilresponse (defined by a spring stiffness) is adopted. Also, some ofthe input parameters remain unchanged throughout the study,with values presented in Table 1.

2. Initial database characteristics

A large database of 43,745 cases was established to capture mostrealistic SCR dynamic behaviours. Dynamic time history analyseswere carried out to calculate (steady-state) values of Max DrTDZ

over a single cycle of motion. All the numerical models have a fine

segmentation along the riser length with refinement in the TDZwhere segments vary from 0.5 m to 2.5 m depending on the SCRconfiguration and the severity of the imposed displacement.

2.1. Selected ranges of the dimensionless groups

The selected ranges of the individual input parameters in SCRdesign are shown in Table 2. The same design criteria as inQuéau et al. [7] were used in the choice of extreme values forthe wall thickness (wt), the content density (qcont) and the horizon-tal tension component (To) to be consistent with industry practices.Design criteria specific to the dynamic cases were also imple-mented between the heave amplitude (H) and the period (T) ofthe imposed vessel motions. These values were establishedthrough numerical experiments using typical SCR configurationsand different wave spectra (from Gulf of Mexico) to cover a widerange of vessel motions in response to calm and to very harshsea states. However, a narrower range of H values are usually thedominant ones for fatigue design purpose.

The ranges of the dimensionless groups are presented in Table 3as deduced from the chosen values of individual input parameters.Particular cases were defined by selecting appropriate combina-tions of dimensionless groups, respecting the design criteria onthe individual input parameters.

2.2. Cases forming the overall database

DoE techniques were used to establish the database in anattempt to capture the boundaries of the input design spaces whileproviding homogeneous coverage over the entire domain. Thedetailed explanation of DoE techniques is well covered in theliterature (e.g. [9,16–19]). In short, DoE is a method that is appliedto gain as much knowledge as possible from ‘experimental’ resultsthrough a limited number of experiments by using various sta-tistical techniques. The methods used in this paper are (i) the fullfactorial approach, consisting in discretising the ranges of the inputdesign parameters in a number of levels and testing the effects ofevery possible combinations of the levels of the input designparameters on the output; and (ii) a quasi-random approach, aim-ing at spreading the cases within the design space.

A total of 12,288 cases were obtained with a full factorialdesign while the remaining cases were obtained through quasi-random techniques. The full factorial design cases were derivedfrom the cases developed for the static study [7]. Additional levelsof H and two levels for T were selected to account for the selecteddesign criteria on the extreme values of T, as presented in Table 2.This led to the following levels, with the superscript ± referring tothe fact that values of those input parameters either side of theextremities of the ranges involved in the design criteria weretested for the various possible ranges of the dependent parameters(e.g. Dz = 1500 m was tested with values of hHO corresponding tothe intervals relevant for both Dz just less than 1500 m(9� 6 hHO 6 17�) and Dz just greater than 1500 m (7� 6 hHO 6 11�)):

� Dz: 6 levels selected: 400 m; 950± m; 1500± m; and 2000 m.� Do: 8 levels selected: 0.1524 m; 0.36± m; 0.46± m; 0.56± m and

0.762 m.� p6: = Do/wt: two levels selected: appropriate minimum and

maximum values with respect to the value of Do.� qcont: two levels selected: empty and full with appropriate con-

tent density value with respect to the value of Do.� To: two levels selected: appropriate minimum and maximum

values determined through the value of hHO with respect tothe value of Dz.� H: eight levels selected: 0.1 m; 1± m; 3.5± m; 5.5± m and 7.5 m.� ks: two levels selected: 11.4 kPa and 228 kPa.

Fig. 1. Flowchart of the automation subroutine for dynamic loading conditions.

(a) (b)

Fig. 2. SCR geometry and key parameters: (a) side view of SCR configuration; and(b) riser cross-section [9].

Table 1Unchanged input parameters.

Input parameter Value

Gravity acceleration, g 9.81 m/s2

Young’s modulus for the SCR, E 2.12E8 kPaWater density, qwater 1.025 te/m3

Steel density, qsteel 7.85 te/m3

Angular position on the SCR circumference, b 0�Angle of the motion relative to the hang-off angle (hHO), Dhm 0�Soil friction coefficient, l 0.5Poisson’s ratio, m 0.293Drag coefficient, CD 1.2Added mass coefficient, CA 1

174 L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185

� T: two levels selected: appropriate minimum value determinedthrough the value of H and a maximum value of 20 s.

As a result, there are two levels for the dimensionless soil stiff-ness (p11) and six levels for the dimensionless water depth (p15),which in turn determine the relevant eight levels for the dimen-sionless displacement amplitude (p2) and outside diameter (p5)and the two levels for the dimensionless tension (p9), subsequentlyleading to the appropriate two levels for the dimensionless dis-placement velocity (p4), outside diameter over wall thickness ratio

(p6) and dimensionless submerged weight (p7). This number ofselected levels corresponds to a total of 12,288 combinations.

The relationships between the input dimensionless groups andMax DrTDZ/E are expected to be more complex for the presentdynamic study than for the previous static study [7] due to theadditional nonlinearities introduced by the hydrodynamic forcesand the dynamic amplifications effects for example. For this reasonit would seem reasonable to define an overall database having atleast the same size as the database established for the static study(i.e. �57,000 cases). However, dynamic analyses of SCRs are muchmore time consuming than static analyses and the resulting sim-ulation files occupy a large amount of disk space (as noted alsoby Xia et al. [4]). The number of additional cases obtained withquasi-random techniques was therefore mostly limited by thecomputational resources available for this study. A total of31,457 additional cases were added using quasi-random

Table 2Selected extreme values of the individual input parameters.

Varied inputparameter

Minimum value Maximum value

Water depth, Dz 400 m 2000 mOutside diameter, Do 0.1524 m 0.762 mWall thickness, wt Do/15 if

0.1524 m 6 Do < 0.36 mDo/9 if0.1524 m 6 Do 6 0.36 m

Do/20 if0.36 m 6 Do < 0.56 m

Do/12 if0.36 m < Do 6 0.56 m

Do/25 if0.56 m 6 Do 6 0.762 m

Do/15 if0.56 m < Do 6 0.762 m

Content density, qcont 0 kg/m3 1025 kg/m3 if0.1524 m 6 Do 6 0.46 m800 kg/m3 if0.46 m < Do 6 0.762 m

Horizontal tensioncomponent, To

Such that: Such that:hHO = 17� if400 m 6 Dz < 950 m

hHO = 20� if400 m 6 Dz 6 950 m

hHO = 9� if950 m 6 Dz < 1500 m

hHO = 17� if950 m < Dz 6 1500 m

hHO = 7� if1500 m 6 Dz 6 2000 m

hHO = 11� if1500 m < Dz 6 2000 m

Heave amplitude, H 0.1 m 7.5 mSoil stiffness, ks 11.4 kPa 228 kPaPeriod of the input

motion, T4 s if 0.1 m 6 H < 1 m 20 s7 s if 1 m 6 H < 3.5 m10 s if3.5 m 6 H < 5.5 m13 s if5.5 m 6 H 6 7.5 m

Table 3Selected ranges of the dimensionless groups for SCR stress analysis.

Varied dimensionless group Minimumvalue (–)

Maximumvalue (–)

Riser displacement amplitude, p2 = H/Dz 5.00E�05 1.88E�02Riser displacement velocity, p4 ¼ H

T

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqsteel=E

p9.62E�07 1.11E�4

Riser outside diameter, p5 = Do/Dz 7.62E�05 1.91E�03Riser outside diameter to wall thickness ratio,

p6 = Do/wt

9 25

Riser unit submerged weight, p7 = p/(E Dz) 3.92E�13 8.07E�11Riser tension, p9 = To/(E Dz2) 5.44E�14 4.19E�11Soil stiffness, p11 = ks/E 5.38E�08 1.08E�06Water depth, p15 = g Dz qsteel/E 1.45E�04 7.26E�04

L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185 175

techniques leading to an overall database of 43,745 cases,representing a total of about four months of calculation (auto-mated pre-processing, running of the simulations and post-pro-cessing of the simulation files) and a hard disk occupation of6 TB, using a specially selected high performance computer witha Microsoft Windows Seven operating system, an Intel Core i7980@3.33 GHz processor, a memory of 24 GB and able to runtwelve cases at a time. This initial database could be expanded infuture work, depending on the performance of the results obtained,as discussed later on in the paper.

An illustration of the values of the dimensionless groups for theentire database is given in Fig. 3a showing the coverage of thedesign space. The figure indicates that the database encompassescases able to test the entire ranges of every dimensionless group.Nonetheless, the output design space does not seem completelywell captured by the database as there seems to be irregular highvalues of Max DrTDZ/E with isolated spot values. This could comefrom the selected combination of dimensionless groups withinthe cases of the database. Results of a more detailed investigationare shown in Fig. 3b where the subspaces created by every possiblepair of dimensionless groups among themselves and with the out-put (Max DrTDZ/E) are represented. The complex shapes are due tothe various selected design criteria. Fig. 3b shows that the sub-spaces created by the pair of dimensionless groups are well

populated. However, the boundaries of the subspaces createdbetween the output and p2, p5, p6 and p11 are rather ill-definedwith several outliers observed. This could indicate that more casesin these targeted areas would be needed to accurately capture theunderlying relationships between dimensionless groups. Despitethese observations, the current database is used to get some firstinsights on the viability and accuracy of the overall approach.Future work could, if necessary, aim at refining the method andthe selected ranges for the dimensionless groups using the knowl-edge gained from the present study.

2.3. Training and testing sets

The database needs to be divided into a training set and a testingset. The training set is used to train the ANN and should contain allthe information that needs to be captured by the approximationwhile the testing set aims at evaluating the interpolation ability ofthe defined approximation. Accepted practice is to use about twothirds of a database for training and the remaining cases for testing[12,20]. It was decided to use all of the cases of the full factorialdesign for training and select additional cases in order to get similarstatistical properties between the training and the testing set. Thisis called a statistically consistent method and was used successfullyfor the static study [7] and in other fields of application (e.g. [12]). Itis important that the testing set gives a good representation of theoverall design space to obtain reliable results when evaluating theperformance of the trained ANN (otherwise some potential margin-ally inaccurate areas could be undetected). A trial-and-errorapproach is used to get acceptable statistics for the two sets.

A total of 29,200 cases were used for training and 14,545 fortesting, with the characteristics of both sets summarised inTable 4. Extreme values of the input dimensionless groups are pre-sented for the two sets and differences in average values and stan-dard deviations are mainly small (despite a relatively largedifference in standard deviation for the soil stiffness (p11)).Regarding the output, the statistics of the testing set are similarto those of the training set, although extreme values are not strictlymatched.

3. Development of an approximation of the maximum stressrange using the initial database

3.1. Selected architecture for the approximation

The ANN development tool implemented in modeFRONTIER, asused by Quéau et al. [7] when developing an approximation of thecritical static stress range, is used here by default. It is based on aone-hidden layer Levenberg–Marquardt back-propagation neuralnetwork with a maximum number of neurons in the hidden layerof 100. In the static study by Quéau et al. [7], some of the ranges ofthe dimensionless groups were subdivided into smaller ranges toimprove the level of accuracy of the resulting ANNs. The sameapproach was followed for the dynamic cases here as, at first, a sin-gle ANN was trained for the overall database, but the accuracy ofthe resulting approximation (Approximation 1), was insufficient.

The results of the various attempts are reported in Table 5; theyallow identification of the parts of the design space that need fur-ther attention. At this stage, the selected indicators of performancefor the developed ANNs are: (i) the extreme values of their relativeerrors with the results obtained from OrcaFlex simulations for thetraining and testing sets; and (ii) the proportion of cases within±5% relative errors. Due to the large range of relative errorsobtained with Approximation 1, Approximation 2 was defined bytraining three ANNs for the groups of cases within each of the threeselected ranges of Dz, leading to various ranges for hHO (low:

Fig. 3. Scatter plot of selected cases: (a) Individual dimensionless groups; and (b) pairs of dimensionless groups.

176 L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185

400 m 6 Dz < 950 m, medium: 950 m 6 Dz < 1,500 m and high:1,500 m 6 Dz 6 2,000 m). The cases from the training and testingsets falling within these three intervals of Dz were used to trainand test the three new ANNs.

Most of the input dimensionless groups depend on Dz, which iswhy it was selected for an initial division of the design space.Approximation 2 was an improvement compared withApproximation 1 but yet did not reach sufficient accuracy.Further divisions were added within each of the three groupsdefined for Approximation 2, this time depending on the value of

the riser displacement (p2) to form Approximation 3. Cases wereseparated into two groups at the middle of the appropriate inter-vals of variation of p2 (i.e. lower and upper half of p2 dependingon the range of Dz) leading to six ANNs. An additional division atone-eighth of the appropriate ranges of p2 was made inApproximation 4, which is formed by nine ANNs and has a similarstructure as the final approximation defined for the static cases (‘9-ANNs static approximation’ in Quéau et al. [7]). Approximations 3and 4 showed that, regardless of the value of Dz, the accuracy ofthe fit improved with the increase of p2, but yet the accuracy for

Table 4Characteristics of the overall training and testing databases.

Max DrTDZ/E p2 p4 p5 p6 p7 p9 p11 p15

Training set (29,200 cases)Maximum 7.25E�03 1.88E�02 1.11E�04 1.91E�03 25.00 8.07E�11 4.19E�11 1.09E�06 7.26E�04Minimum 2.58E�06 5.00E�05 9.62E�07 7.62E�05 9.00 3.92E�13 5.44E�14 5.38E�08 1.45E�04Mean 1.06E�03 5.58E�03 5.60E�05 7.76E�04 16.23 2.13E�11 9.37E�12 5.73E�07 3.58E�04Standard deviation 8.67E�04 4.92E�03 3.15E�05 5.56E�04 4.22 2.20E�11 1.09E�11 3.98E�07 1.94E�04

Testing set (14,545 cases): Relative difference with training setMaximum �13.55% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.03% 0.00%Minimum 8.99% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Mean 3.63% 6.09% 4.69% �0.01% 1.12% �4.38% �4.85% �1.38% �0.22%Standard deviation �7.01% 2.93% �10.99% �0.64% �6.97% �1.47% �2.00% �20.10% �1.41%

Table 5Performance of the various attempted approximations of Max DrTDZ/E.

Approximation Comment NumberANN

Region of thedesign space

Training Testing

Relative differencewith OrcaFlex results

Proportionof caseswith errorswithin ±5%(%)

Relative differencewith OrcaFlex results

Proportionof caseswith errorswithin ±5%(%)

Maximum(%)

Minimum(%)

Maximum(%)

Minimum(%)

1 Overall database 1 Entire design space 2015 �1850 72 952 �487 79

2 Dz split 3 ways 3 Low Dz (400 m 6 Dz < 950 m) 1001 �622 83 503 �73 86Medium Dz (950 m 6 Dz < 1500 m) 712 �447 73 931 �131 76High Dz (1500 m 6 Dz 6 2000 m) 830 �623 74 415 �273 76

3 Dz split 3 ways,subsequent p2

split 2 ways

6 Low Dz, Lower half p2 1099 �701 75 366 �72 77

Low Dz, Higher half p2 4 �5 100 32 �14 99Medium Dz, Lower half p2 757 �738 72 507 �133 70Medium Dz, Higher half p2 14 �14 95 56 �57 64High Dz, Lower half p2 404 �195 81 178 �233 78High Dz, Higher half p2 38 �29 96 30 �35 79

4 Dz split 3 ways,subsequent p2

split 3 ways

9 Low Dz, Lower 1/8 p2 694 �517 58 647 �195 23

Low Dz, Remaining lower half p2 37 �35 90 53 �43 87Low Dz, Higher half p2 4 �5 100 32 �14 99Medium Dz, Lower 1/8 p2 185 �150 79 1281 �355 29Medium Dz, Remaining lower half p2 22 �23 89 44 �35 76Medium Dz, Higher half p2 14 �14 95 56 �57 64High Dz, Lower 1/8 p2 156 �90 81 293 �265 31High Dz, Remaining lower half p2 13 �36 96 38 �31 77High Dz, Higher half p2 38 �29 96 30 �35 79

5 Dz split 3 ways,subsequent p4

split 2 ways

6 Low Dz, Lower half p4 421 �508 66 540 �122 38

Low Dz, Higher half p4 29 �19 97 25 �27 96Medium Dz, Lower half p4 105 �137 85 2177 �451 36Medium Dz, Higher half p4 30 �39 86 59 �35 82High Dz, Lower half p4 166 �73 87 291 �141 41High Dz, Higher half p4 26 �31 91 32 �35 86

6 Dz split 3 ways,subsequent p4

split 3 ways

9 Low Dz, Lower 1/8 p4 282 �254 69 1496 �337 9

Low Dz, Remaining lower half p4 5 �5 100 159 �107 28Low Dz, Higher half p4 29 �19 97 25 �27 96Medium Dz, Lower 1/8 p4 66 �42 92 1751 �1426 8Medium Dz, Remaining lower half p4 3 �2 100 185 �303 11Medium Dz, Higher half p4 30 �39 86 59 �35 82High Dz, Lower 1/8 p4 50 �60 92 789 �467 13High Dz, Remaining lower half p4 5 �1 100 77 �133 21High Dz, Higher half p4 26 �31 91 32 �35 86

L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185 177

the low values of p2 was insufficient. Further subdivisions on p2

values where attempted but did not lead to better results.To explore the performance of other subdivisions, another

strategy was used for Approximations 5 and 6, which have the

same structure as Approximations 3 and 4 respectively but withsubdivisions based on the value of the imposed displacementvelocity (p4) due to its fundamental role for the dynamic cases.This approach did not offer better performance and led to over

178 L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185

fitting some areas of the design space. Over fitting happens whenan ANN can accurately predict the results for the training set butcannot interpolate correctly; it is not able to generalise from thetrends given by the training set and thus offers poor performancefor the testing set. This could arise from the use of too many neu-rons in the hidden layer or from poor quality of the data used fortraining.

Both of the more advanced approximations using nine ANNstherefore needed further refinements. It was chosen to try to refineApproximation 4 in order to get a similar architecture as for theapproximation developed for the static cases by Quéau et al. [7].

3.2. Refinement of ANNs inherent to Approximation 4

In view of the performance of Approximation 4 in Table 5, thesubspaces at the lowest end of the riser displacement (p2) intervalsfor each range of Dz were the prime target for refinement as theypresented the poorest performances. MATLAB [21] was used forrefinement as it offers more freedom than modeFRONTIER in thechoice of ANN type and architecture. For instance, using MATLABit is possible to test the effect of an addition of one or several hid-den layers to the ANN architecture, a feature not available inmodeFRONTIER. The number of neurons was varied following atrial-and-error approach to find an optimum solution. The numberof hidden layers was also varied but it was shown not to improvethe performance of the approximations.

The best results were achieved by using only one hidden layer(as used by default originally), 50 neurons for the ANNs used for

Fig. 4. Structure of the 9-ANN

the lowest intervals of p2 for the low and medium range of Dzand 20 neurons for the lowest intervals of p2 for the high rangeof Dz. However, it was necessary to exclude the very low p2 values(representing very small motion amplitudes as a proportion of thewater depth) from the range of application of the ANNs where highdiscrepancies between OrcaFlex and the approximation werefound in the training and in the testing set. This was the case forp2 6 6.88E�4 for the low range of Dz, p2 6 1.89E�4 for the med-ium range of Dz and for p2 6 1.27E�4 for the high range of Dz.(These values correspond to the lowest 1/32nd of the overallselected range of variation for p2 for the low range of Dz and thelowest 1/64th of the overall selected range of variation for p2 forthe medium and high ranges of Dz.) These values were howeverused for training the ANNs as excluding them from the trainingset decreased the performances on the rest of the selected rangesof p2.

The refined Approximation 4 has the architecture presented inFig. 4 and introduces the notations for the ANNs. ANN1 corre-sponds to the ANN having 50 neurons in its hidden layer and devel-oped for the region of the design space indicated as ‘Low Dz, Lower1/8 p2’ in Table 5. ANN2 and ANN3 both have the default numberof neurons in their hidden layer (i.e. 100) and correspond respec-tively to the regions of the design space indicated as ‘Low Dz,Remaining lower half p2’ and ‘Low Dz, Higher half p2’ in Table 5.ANN4, ANN5 and ANN6 are the homologues of ANN1, ANN2 andANN3 respectively in the selected medium range of water depthas are ANN7, ANN8 and ANN9 in the selected high range of waterdepth. When using this approximation, it takes only a couple of

s dynamic approximation.

Table 7Performance of the final approximation of Max DrTDZ/E on the overall design space(accounting for the introduced limits on the value of p2).

L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185 179

minutes to get an evaluation of Max DrTDZ/E for the entire database(by contrast with the four months of calculation taken for the cre-ation of the database with numerical analyses).

Training set Testing set

1-Correlation, 1 � r 1.12E�03 4.26E�03Relative difference with

OrcaFlex results:Maximum 124.83% 143.19%Minimum �47.72% �57.38%RMSE 0.20% 0.40%MAE 2.14% 3.32%

Proportion of cases witherrors within:

±5% 90.54% 80.40%±10% 97.02% 93.00%±15% 98.60% 96.67%

3.3. Performances of Approximation 4

The individual performances of each of the inherent ANNs areillustrated in Table 6, using notations introduced in Fig. 4 whilethe performance of the overall approximation is presented inTable 7. The root mean squared errors (RMSE) and mean absoluteerrors (MAE) are evaluated too as a further indication of the perfor-mance of the ANNs. For the testing set, the RMSE and MAE are lowand there are �96% and �80% of the cases within ±15% and ±5%relative difference with OrcaFlex results respectively. The definedANNs forming Approximation 4 therefore provide a good basisfor a first approximation as part of this pilot study since, despitesome marginal high values of relative differences within someANNs, a large proportion of the cases of the database are withina negligible range of error.

4. Refinement of the approximation for part of the design spaceby expanding the database

The focus is now brought to part of the design space wheresome of the marginally high relative differences between esti-mated and OrcaFlex stress range results were observed.Additional cases were added to the initial database in this part ofthe design space to explore the effect of refining the database onimprovement of the ANN approximation, prior to testing its accu-racy on fatigue predictions.

The refinement is applied only on the part of the designspace initially targeted by ANN4 (i.e. without the limit on p2),which corresponds to 950 m 6Dz < 1500 m, 9� 6 hHO 6 17� and6.67E�5 6 p2 6 1.05E�3. This is an arbitrary choice based on (a)the fact that Dz was set to 982 m in the SCR base case used bythe authors [10,22], corresponding to the medium range ofselected Dz and (b) that, within this range of Dz, the areacorresponding to 6.67E�5 6 p2 6 1.05E�3 presented the least

Table 6Performance of the ANNs forming the final approximation of Max DrTDZ/E.

ANN1a ANN2 ANN3 ANN4a

Training set1-Correlation, 1 � r 1.37E�02 2.22E�03 1.23E�04 2.44E�Relative

differencewithOrcaFlexresults:

Maximum 124.83% 36.81% 4.33% 88.01%Minimum �40.44% �34.73% �4.84% �47.72%RMSE 1.61% 0.15% 0.00% 0.43%MAE 7.11% 2.31% 0.39% 3.97%

Proportion ofcases witherrorswithin:

±5% 60.68% 89.64% 100.00% 74.30%±10% 80.06% 96.74% 100.00% 91.37%±15% 87.45% 98.71% 100.00% 97.36%

Testing set1-Correlation, 1 � r 3.63E�02 4.51E�03 1.79E�03 1.83E�Relative

differencewithOrcaFlexresults:

Maximum 143.19% 52.55% 31.97% 63.29%Minimum �40.59% �42.61% �13.74% �52.23%RMSE 2.32% 0.20% 0.02% 1.02%MAE 9.25% 2.54% 0.67% 6.81%

Proportion ofcases witherrorswithin:

±5% 47.91% 87.12% 98.77% 51.03%±10% 70.70% 96.02% 99.63% 78.24%±15% 81.28% 98.70% 99.89% 89.68%

a With introduced limit on the value of p2.

accurate results. The initial database comprised 2338 cases in thispart of the design space.

4.1. Detailed analysis of the initial database on the selected part of thedesign space

Two strategies were implemented to select the additional casesand extend the database. On the one hand the training set size wasincreased to capture the relationships better between the inputdimensionless groups and the output; on the other hand, morecases were added to the testing set to match the statistics of thetraining set defined in this part of the design space, and improvethe assessment of the interpolation ability of the trained ANN.(The statistically consistent approach was used for the overalltraining and testing sets and therefore does not necessarily guaran-tee similar statistical consistency for the training and testing setsformed by the subsequent divisions of the design space, dependingon the value of the riser displacement (p2) and water depth (p15).)

Plots shown in Fig. 5 were used to assist the selection of theadditional cases. The blue parts of Fig. 5 represent the cases ofthe current database in this reduced part of the overall designspace whereas the red parts illustrate the additional cases, as dis-cussed in Section 4.2. Plots are similar to those in Fig. 3, with thediagrams in Fig. 5a representing values of either the output or aparticular input dimensionless group for each case, and in Fig. 5bthe output value for each value of the various input dimensionless

ANN5 ANN6 ANN7a ANN8 ANN9

03 1.58E�03 8.57E�04 1.12E�03 2.10E�03 2.48E�0322.42% 13.96% 27.80% 13.27% 37.54%�22.72% �14.37% �35.25% �35.98% �28.53%

0.13% 0.06% 0.23% 0.06% 0.07%2.46% 1.82% 3.39% 1.58% 1.72%

88.59% 94.92% 78.23% 96.13% 95.88%97.79% 99.73% 95.81% 99.52% 99.19%99.18% 99.73% 98.63% 99.82% 99.69%

02 9.94E�03 1.34E�02 7.31E�03 8.56E�03 9.78E�0344.14% 55.80% 66.21% 37.85% 29.71%�35.44% �57.38% �32.37% �30.63% �35.27%

0.30% 0.67% 0.67% 0.28% 0.29%3.70% 5.39% 5.06% 3.61% 3.53%

76.14% 63.50% 65.66% 77.17% 78.65%94.04% 86.99% 87.01% 94.38% 93.69%97.59% 93.78% 93.74% 97.85% 97.51%

(a)

(b)

Fig. 5. Scatter plot of the cases selected initially (in blue) and the additional cases (in red) in the area targeted by ANN4: (a) Individual dimensionless groups; and (b) pairs ofdimensionless groups. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

180 L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185

groups, or pairs of values of different input dimensionless groups,within the reduced part of the overall design space.

Fig. 5a shows that the ranges of the input dimensionless groupsare not always covered homogenously, for instance more casessimulating values within the higher end of the ranges for the risersubmerged weight (p7) and the riser tension (p9) could be added

for a better accuracy since isolated high values of p7 and p9 weredetected. It is (trivially) an easier task to map the input designspace appropriately than the output space and Fig. 5b gives a fur-ther indication on the location of the ‘gaps’ within the part of thedesign space under study. For instance, the soil stiffness (p11) iscompletely independent from any of the other input dimensionless

Table 8Characteristics of the initial training and testing database in the area targeted by ANN4 (no limit on the value of p2).

Max DrTDZ/E p2 p4 p5 p6 p7 p9 p11 p15

Training set (1,727 cases): Maximum 1.86E�03 1.04E�03 4.76E�05 8.02E�04 25.00 3.40E�11 1.40E�11 1.08E�06 5.45E�04Minimum 3.09E�06 6.67E�05 9.62E�07 1.02E�04 9.00 5.23E�13 9.69E�14 5.38E�08 3.45E�04Mean 2.62E�04 5.13E�04 1.63E�05 3.79E�04 16.08 7.88E�12 2.37E�12 5.57E�07 4.62E�04Standard deviation 2.62E�04 3.53E�04 1.57E�05 1.66E�04 4.80 6.11E�12 2.11E�12 4.66E�07 8.96E�05

Testing set (611 cases)Relative differencewith training set:

Maximum �41.65% 0.07% 1.01% �1.43% 0.00% �3.76% �33.11% 0.71% �0.05%Minimum �9.16% 10.93% 0.00% 2.16% 0.00% 26.02% 119.11% 0.00% 0.01%Mean �1.41% 7.37% 3.69% 2.57% 1.65% �1.10% �2.50% 3.32% �2.21%Standard deviation �47.07% �21.51% �35.28% �2.04% �10.48% �5.11% �16.87% �33.14% �36.72%

Table 9Characteristics of the refined training and testing database in the area targeted by ANN4 (no limit on the value of p2).

Max DrTDZ/E p2 p4 p5 p6 p7 p9 p11 p15

Training set (5,797 cases): Maximum 1.86E�03 1.05E�03 4.81E�05 8.02E�04 25.00 3.40E�11 1.40E�11 1.08E�06 5.45E�04Minimum 3.09E�06 6.67E�05 9.62E�07 1.02E�04 9.00 5.22E�13 9.69E�14 5.38E�08 3.45E�04Mean 2.06E�04 5.00E�04 1.27E�05 5.30E�04 16.55 1.19E�11 4.29E�12 7.43E�07 4.35E�04Standard deviation 2.09E�04 4.71E�04 1.38E�05 2.18E�04 4.68 8.20E�12 3.32E�12 4.00E�07 8.21E�05

Testing set (2,580 cases)Relative differencewith training set:

Maximum �39.39% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.71% 0.00%Minimum �9.16% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%Mean �2.47% 2.05% �1.02% 3.83% 1.07% 4.15% 4.95% 2.09% �1.20%Standard deviation �15.83% 1.32% �9.82% �2.67% �3.67% �0.20% �0.26% �12.19% �12.40%

Table 10Performance of ANN4⁄ (with the same limit on the value of p2 than for ANN4).

L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185 181

group, which means that, ideally, the scatters should cover a rect-angular shaped area in each of the subplots representing the choiceof soil stiffness in conjunction with another input dimensionlessgroup. Additional cases in the highest end of the ranges of the risersubmerged weight (p7) and the riser tension (p9) for the entirerange of soil stiffness (p11), or in the highest end of the ranges ofthe riser velocity (p4) and the riser outside diameter (p5) for theentire range of soil stiffness, could hence be defined to enhanceperformance. Similarly, the riser velocity is independent from allthe other input dimensionless groups with the exception of theriser displacement (p2) and therefore the subspaces formed bythe level of the riser velocity and any other dimensionless groups(other than p2) should have a rectangular shaped area too toimprove the quality of the database; and so on by consideringthe other pairs of dimensionless groups. These observations pro-vide a first basis for the selection of additional cases.

The quality of the training and testing set in this part of thedesign space was then investigated to refine further the databaseand the performance of the subsequent approximation. A detailedanalysis of the cases from the training and testing databases forthe selected part of the design space is performed in Table 8.The table shows that 1727 cases were selected for training and611 for testing. The statistics of the training and testing setsare compared, revealing target areas for improvements. Forexample, the ranges of input dimensionless groups covered bythe cases from the testing set do not always span over the entireranges of the cases from the training set. This is observed mostevidently for the riser submerged weight (p7) and the riser ten-sion (p9).

ANN4⁄ ANN4

Refinedtraining set

Refinedtesting set

Refinedtesting set

1-Correlation, 1 � r 1.96E�03 5.85E�03 8.54E�02Relative

differencewith OrcaFlexresults:

Maximum 207.86% 129.31% 314.42%Minimum �45.02% �61.48% �86.15%RMSE 0.68% 0.94% 7.07%MAE 3.96% 5.10% 15.51%

Proportion ofcases witherrors within:

±5% 77.75% 71.84% 34.01%±10% 92.40% 88.04% 56.18%±15% 95.99% 93.41% 68.48%

4.2. Improved database characteristics

The original database of 2338 cases in the selected part of thedesign space was extended to 8377 cases, with 5797 for trainingand 2580 for testing. The additional cases were established basedon the observed gaps in the design space with Fig. 5 showing thescatter plot of the values of the input dimensionless groups andoutput for the extended database. In Fig. 5b, the cases of the initialdatabase (in blue) are represented on top of the cases of the

extended database (in red) to highlight the gaps that the new casesare now filling.

The allocation of cases to the training or the testing set was per-formed following a trial-and-error approach aiming to obtain simi-lar statistics between both improved sets, as per the approachapplied for the overall design space. Table 9 summarises the resultsobtained with the larger database in the selected part of the designspace. The improved training and testing set now have similarstatistics, especially for the input design space, and have the poten-tial to improve further the quality of the proposed approximation.

4.3. New approximation on this part of the design space

A new ANN, referred to as ANN4⁄, was trained and tested usingthe improved training and testing sets. As for ANN4, it was trainedin MATLAB and comprised one hidden layer with 50 neurons. Thelimit on the value of the riser displacement (p2) was re-introducedat this stage to be consistent with ANN4 allowing their perfor-mances to be compared appropriately. Only cases with values ofp2 higher than 1.89E�4 (and less than 1.05E�3) are therefore con-sidered here. The performance of ANN4⁄ is assessed using theresults shown in Table 10, where the new testing set was also usedwith ANN4 to enhance comparison between ANN4 and ANN4⁄.

The quality of the results shown in Table 10 for ANN4 hasdecreased compared with the results presented in Table 6, and

Table 11SCR base cases characteristics.

Input parameter/dimensionless group BC1 BC2 BC3 BC4 BC5

Do (m) 0.228 (�9 in) 0.4 (�15.7 in) 0.7 (�27.6 in) 0.6 (�23.6 in) 0.3 (�11.8 in)wt (m) 0.025 (�1 in) 0.027 (�1.1 in) 0.035 (�1.4 in) 0.024 (�1 in) 0.025 (�1 in)p (kN/m) 0.82 2.09 4.21 0.50 0.95Dz (m) 982 1250 1450 1150 1350hHO (�) 9.8 13 17 15 11ks (kPa) 22.8 100 200 50 150qcont (kg/m3) 0 1025 800 0 0To (kN) 164 759 2520 201 303p5 = Do/Dz 2.32E�04 3.20E�04 4.83E�04 5.22E�04 2.22E�04p6 = Do/wt 9.12 15.00 20.00 25.00 12.00p7 = p/(E Dz) 3.92E�12 7.90E�12 1.37E�11 2.06E�12 3.33E�12p9 = To_catenary/(E Dz2) 8.05E�13 2.29E�12 5.65E�12 7.18E�13 7.84E�13p11 = ks/E 1.08E�07 4.72E�07 9.43E�07 2.36E�07 7.08E�07p15 = g Dz qsteel/E 3.57E�04 4.54E�04 5.27E�04 4.18E�04 4.90E�04

182 L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185

Table 10 shows that some high differences with OrcaFlex results onthe value of Max DrTDZ/E were not captured when using the initialtesting set. This emphasises the significance of the choice of casesfor the testing set.

ANN4⁄ is indeed found to be a better approximation than ANN4,offering better correlation, reduced interval of relative differenceand higher number of cases within low range of relative differenceswith OrcaFlex results, confirming that the steps undertaken inSections 4.1 and 4.2 have contributed to refining the quality ofthe approximation in this part of the design space.

The current best approximation, namely ‘‘9-ANNs dynamicapproximation’’, has therefore the structure presented in Fig. 4with ANN4⁄ in place of ANN41. Even though the accuracy of the9-ANNs dynamic approximation does not reach the original criterionof errors within ±5% of errors for the entire design space, the benefitsof the approach for practical purposes (i.e. fatigue life predictions) isinvestigated next.

Table 12Loading conditions for the case studies.

Load case(LC)

Waveheight(m)

HeaveamplitudeH (m)

PeriodT (s)

Number ofoccurrencefor 20 years

LC1 1 0.004a 3 156,673,082LC2 1 0.15 8 10,946,661LC3 1 0.25 13 1,132,827LC4 1 0.39 18 305,883LC5 3 0.44 8 2,795,859LC6 3 0.76 13 130,073LC7 3 1.17 18 16,564LC8 8 1.17 8 68,228LC9 13 1.91 8 2073LC10 8 2.02 13 7409LC11 8 3.12 18 607LC12 13 3.29 13 647LC13 18 4.55 13 92

5. Application of the framework in fatigue design: case studies

The current best approximation (9-ANNs dynamic approx-imation) is now applied on a series of case studies to assess its accu-racy for prediction of fatigue life using a deterministic approach (asappropriate for structural systems presenting nonlinearities [23]).In light of the refinements performed in Section 4, this sectionfocuses on the part of the design space corresponding to theselected medium range of water depth (i.e. 950 m 6 Dz < 1500 m).

5.1. SCR configurations and loading conditions

The first case study, namely Base Case 1 (BC1), is performedwith the usual SCR base case used by the authors (as per BC1 inQuéau et al. [10]). Its input parameters and corresponding dimen-sionless groups are shown in Table 11. The base case is derivedfrom an in-service SCR connected to a semi-submersible in theGulf of Mexico (GoM). Four additional base cases, namely BC2,BC3, BC4 and BC5, were defined in order to test the accuracy ofthe 9-ANNs dynamic approximation for different riser setups (i.e.input data from different areas of the design space). The inputparameters and corresponding dimensionless groups of these basecases are also presented in Table 11. They span over the entiredesign ranges of Do, Dz, hHO, ks, qcont and p6, within the refined areastudied in Section 4.

In this deterministic fatigue design, a series of 15 load cases(LCs) characterised by a sinusoidal tangential heave motion (of

1 The 9-ANNs dynamic approximation can be provided as a standalone applicationby contacting the corresponding author.

amplitude H and period T) are applied to these SCRs in the aimof quantifying the difference in evaluation of the fatigue lifethrough conventional numerical time domain analyses (usingOrcaFlex) and through the proposed 9-ANNs dynamic approx-imation for estimation of the maximum stress range. The detailsof the LCs and corresponding values of the riser displacement(p2) and velocity (p4) are summarised in Table 12 and Table 13respectively. These LCs were derived from a simplified wave scat-ter table for GoM with the wave heights, periods and number ofoccurrences presented in Table 12. The values of the correspondingheave amplitudes were assessed based on RAO (response ampli-tude operator) tables of the vessel. The number of occurrence ofthese waves for 20 years and the waves periods are reported inTable 12 from which a total exposure time of 602,861,328 s canbe calculated. These 15 waves thus represent over 95% of thewaves occurring in the GoM over a 20 year period (i.e.630,720,000 s).

5.2. Fatigue life evaluation

The dynamic time history analyses were carried out usingOrcaFlex to assess the value of Max DrTDZ for each of the LCs.The results are compared with the estimations from the 9-ANNsdynamic approximation in Table 14. Some of the first LCs corre-spond to small displacement amplitudes (p2) which are outsidethe selected ranges and therefore ANN4⁄ was used in extrapolationfor these LCs as indicated in Table 14. For the LCs within theselected range of application of the 9-ANNs dynamic

LC14 13 5.06 18 47LC15 18 7.01 18 4

a Value of H outside of the selected range of displacement.

Table 13Values of normalised displacement (p2) and velocity (p4) in the case studies.

Load case (LC) p2 = H/Dz p4=HT

ffiffiffiffiffiffiffiffiqsteel

E

qBC1 BC2 BC3 BC4 BC5

LC1 4.39E�06 3.45E�06 2.97E�06 3.75E�06 3.19E�06 2.76E�07LC2 1.49E�04 1.17E�04 1.01E�04 1.27E�04 1.09E�04 3.53E�06LC3 2.58E�04 2.02E�04 1.75E�04 2.20E�04 1.87E�04 3.75E�06LC4 3.97E�04 3.12E�04 2.69E�04 3.39E�04 2.89E�04 4.16E�06LC5 4.48E�04 3.52E�04 3.03E�04 3.82E�04 3.26E�04 1.06E�05LC6 7.73E�04 6.07E�04 5.24E�04 6.60E�04 5.62E�04 1.12E�05LC7 1.19E�03 9.35E�04 8.06E�04 1.02E�03 8.66E�04 1.25E�05LC8 1.19E�03 9.38E�04 8.09E�04 1.02E�03 8.69E�04 2.82E�05LC9 1.94E�03 1.52E�03 1.31E�03 1.66E�03 1.41E�03 4.58E�05LC10 2.06E�03 1.62E�03 1.40E�03 1.76E�03 1.50E�03 3.00E�05LC11 3.17E�03 2.49E�03 2.15E�03 2.71E�03 2.31E�03 3.33E�05LC12 3.35E�03 2.63E�03 2.27E�03 2.86E�03 2.44E�03 4.87E�05LC13 4.64E�03 3.64E�03 3.14E�03 3.96E�03 3.37E�03 6.74E�05LC14 5.16E�03 4.05E�03 3.49E�03 4.40E�03 3.75E�03 5.41E�05LC15 7.14E�03 5.61E�03 4.84E�03 6.10E�03 5.19E�03 7.50E�05

Table 14Comparison of critical stress ranges and damage results from OrcaFlex and 9-ANNs dynamic approximation.

Load case(LC)

Max DrTDZ Damage (per year) Comment

OrcaFlexresults (kPa)

Relative difference of approximatedresults with OrcaFlex (%)

From OrcaFlexresults

Relative difference of approximatedresults with OrcaFlex (%)

BC1LC1 3865 �92 1.75E�06 �100 Use of ANN4⁄ in extrapolationLC2 12,428 �10 4.21E�05 �41 Use of ANN4⁄ in extrapolationLC3 12,562 8 4.59E�06 49 ANN4⁄

LC4 19,899 �18 1.24E�05 �62 ANN4⁄

LC5 35,733 0 2.11E�03 1 ANN4⁄

LC6 43,023 �5 2.48E�04 �21 ANN4⁄

LC7 54,723 �1 1.05E�04 �3 ANN5LC8 78,648 0 2.66E�03 0 ANN5LC9 113,481 �3 2.73E�04 �9 ANN5LC10 92,026 �4 5.20E�04 �11 ANN5LC11 105,007 �11 6.33E�05 �30 ANN5LC12 129,713 �9 1.27E�04 �26 ANN5LC13 203,586 17 6.99E�05 60 ANN6LC14 158,484 22 1.69E�05 81 ANN6LC15 303,366 12 1.01E�05 39 ANN6

BC2LC1 3179 �9 6.60E�07 �37 Use of ANN4⁄ in extrapolationLC2 15,820 �2 1.41E�04 �8 Use of ANN4⁄ in extrapolationLC3 12,705 40 4.86E�06 441 ANN4⁄

LC4 21,464 �1 1.81E�05 �5 ANN4⁄

LC5 43,809 �2 5.85E�03 �12 ANN4⁄

LC6 46,000 5 3.47E�04 25 ANN4⁄

LC7 57,842 �2 1.39E�04 �11 ANN4⁄

LC8 86,502 1 3.98E�03 4 ANN4⁄

LC9 123,219 0 3.49E�04 �1 ANN5LC10 95,801 1 5.87E�04 2 ANN5LC11 106,566 1 6.62E�05 3 ANN5LC12 130,580 1 1.30E�04 2 ANN5LC13 179,976 2 4.83E�05 7 ANN5LC14 146,000 7 1.32E�05 21 ANN6LC15 239,519 1 4.95E�06 4 ANN6

BC3LC1 6226 0 1.90E�05 1 Use of ANN4⁄ in extrapolationLC2 18,709 2 3.25E�04 9 Use of ANN4⁄ in extrapolationLC3 25,505 �28 1.58E�04 �81 Use of ANN4⁄ in extrapolationLC4 13,878 44 2.04E�06 516 ANN4⁄

LC5 45,639 5 7.17E�03 28 ANN4⁄

LC6 53,654 �8 7.49E�04 �35 ANN4⁄

LC7 54,921 �1 1.07E�04 �3 ANN4⁄

LC8 89,338 �1 4.38E�03 �3 ANN4⁄

LC9 128,822 �15 3.99E�04 �38 ANN5LC10 98,707 �13 6.42E�04 �33 ANN5LC11 108,790 �8 7.04E�05 �22 ANN5LC12 136,854 �6 1.49E�04 �17 ANN5LC13 188,674 �1 5.57E�05 �2 ANN5

(continued on next page)

L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185 183

Table 14 (continued)

Load case(LC)

Max DrTDZ Damage (per year) Comment

OrcaFlexresults (kPa)

Relative difference of approximatedresults with OrcaFlex (%)

From OrcaFlexresults

Relative difference of approximatedresults with OrcaFlex (%)

LC14 149,770 2 1.42E�05 5 ANN5LC15 216,207 3 3.64E�06 9 ANN6

BC4LC1 2298 214 1.30E�07 30,250 Use of ANN4⁄ in extrapolationLC2 12,212 15 3.85E�05 97 Use of ANN4⁄ in extrapolationLC3 15,962 10 1.52E�05 63 ANN4⁄

LC4 18,561 21 8.73E�06 155 ANN4⁄

LC5 34,516 9 1.77E�03 52 ANN4⁄

LC6 42,736 6 2.40E�04 32 ANN4⁄

LC7 51,885 7 8.07E�05 41 ANN4⁄

LC8 73,211 �1 1.86E�03 �6 ANN4⁄

LC9 120,783 �7 3.29E�04 �19 ANN5LC10 90,457 �2 4.94E�04 �5 ANN5LC11 109,975 0 7.27E�05 1 ANN5LC12 140,176 1 1.61E�04 5 ANN5LC13 223,958 13 9.31E�05 45 ANN5LC14 184,855 12 2.67E�05 39 ANN6LC15 323,508 �3 1.22E�05 �7 ANN6

BC5LC1 2701 2 2.92E�07 12 Use of ANN4⁄ in extrapolationLC2 11,466 21 2.81E�05 156 Use of ANN4⁄ in extrapolationLC3 16,705 �8 1.91E�05 �33 Use of ANN4⁄ in extrapolationLC4 15,529 22 3.58E�06 173 ANN4⁄

LC5 40,045 �1 3.73E�03 �4 ANN4⁄

LC6 47,069 �1 3.89E�04 �7 ANN4⁄

LC7 54,763 �1 1.06E�04 �6 ANN4⁄

LC8 78,912 4 2.71E�03 21 ANN4⁄

LC9 117,096 3 3.00E�04 8 ANN5LC10 88,040 3 4.55E�04 10 ANN5LC11 97,910 0 5.13E�05 �1 ANN5LC12 125,179 0 1.14E�04 �1 ANN5LC13 212,839 1 7.99E�05 2 ANN5LC14 153,670 0 1.54E�05 0 ANN5LC15 320,623 �2 1.19E�05 �5 ANN6

Table 15Comparison of fatigue life results from OrcaFlex and 9-ANNs dynamic approximation.

Fatigue life

OrcaFlexresults (yrs)

Relative difference of approximatedresults with OrcaFlex (%)

BC1 160 2.4BC2 86 3.8BC3 70 �7.5BC4 192 �15.8BC5 125 �5.8

184 L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185

approximation, the differences in stress range are small overall,ranging mainly from 0% to 10% relative difference with marginalhigher differences of up to 45% (e.g. LC4 in BC3). Extrapolatedstress range results vary in accuracy from 0% (for LC1 of BC3) rela-tive difference with OrcaFlex results up to 214% (for LC1 of BC4).

Based on the number of wave occurrences presented inTable 12, the fatigue damage created by each of the LCs was calcu-lated by using the D-type S-N curve for seawater below [24]

Log10ðNÞ¼11:764�3� log10ðMaxDrTDZÞ for N6106 cycles

Log10ðNÞ¼15:606�5� log10ðMaxDrTDZÞ for N>106 cyclesð2Þ

Due to the nonlinear relationship between the stress range and thedamage introduced by the number of occurrences of the wave andnumber of allowable cycles, for some cases small relative differ-ences on Max DrTDZ can lead to high differences on damage foran individual load case. This is observed, for instance, for LC6 inBC1 or LC5 in BC4 where only �5% and 9% relative differences inMax DrTDZ respectively lead to �21% and 52% relative differencesin the damage. As the fatigue life is the inverse of the sum of theindividual damage contribution for each LC, small errors in the esti-mated Max DrTDZ for a load case that has a high contribution to theglobal damage can have a high impact on the estimated fatigue life(e.g. for LC5 in BC4) and vice versa for a load case that does not con-tribute much to the global damage (e.g. for LC1 in BC4). However,the positive and negative errors in damage contribution from eachload case tend to balance out so that the total damage estimatedshows much smaller error than the individual components. In addi-tion, the relative accuracy for the load cases that dominate fatigue

damage (typically LC5 to LC10) is generally much better than forthe small amplitude load cases.

The fatigue lives of the five BCs are calculated and results aresummarised in Table 15. The estimated fatigue lives using 9-ANNs dynamic approximation are all reasonably close to those cal-culated using conventional time history analysis using OrcaFlexdirectly, thus demonstrating the benefits and usefulness of theANN method. All BCs have an estimated fatigue life within 16%relative differences with the results derived from OrcaFlex cal-culations, with most BCs within 10% relative difference. Based onthe damage calculated from OrcaFlex stress range results, LC8and LC5 correspond to the waves having the highest contributionsto the global damage for all BCs. The damage for LC5 was not aswell estimated in BC4 than in the other base cases (with a notableoverestimation), which is why the fatigue life of BC4 approximatedby the ANN approach is less than the fatigue life calculated basedon OrcaFlex results by a larger amount than for the other basecases.

L.M. Quéau et al. / Engineering Structures 92 (2015) 172–185 185

Using the 9-ANNs dynamic approximation, it takes less than1 min of calculation to obtain the results presented in Table 15(since the time taken to develop the approximation by trainingthe ANNs does not impact the final application time). Although itis hard to estimate accurately the time necessary to obtain equiva-lent results from OrcaFlex, since it will vary between users anddepending on the available computer resources, it could take abouta day to pre-process the numerical models, simulate them andpost-process the results. The ANN approach therefore providesexcellent efficiency for estimation of fatigue life for different SCRconfigurations.

6. Conclusions

The study presented in the paper has shown how artificial neu-ral networks (ANNs) can be used effectively to provide a simplifiedfatigue design approach for the touchdown zone of SCRs.

A large database of SCR configurations subjected to dynamicloading was generated using OrcaFlex software to provide supportfor the training of a response surface, using ANNs. The size of thedatabase was limited by the time taken to run each of the casesin OrcaFlex. Design of experiment methods were applied whengenerating the database to capture the complex relationshipsbetween the eight selected input dimensionless groups and theoutput (Max DrTDZ). Different ANN configurations were testedand a framework comprising nine ANNs was selected since itwas able to estimate the critical stress range results for the major-ity of the cases of the database within ± 5% relative difference withnumerical results. A refinement of the quality of the database wasapplied for part of the design space to explore the effect of the sizeof training and testing sets and the choice of cases within thesesets on the performance of the approximation. The current bestANN approximation (accounting for the refinement of the databasein part of the design space), referred to as ‘9-ANNs dynamicapproximation’, was used to assess the fatigue lives of exampleSCRs under a selected small wave scatter diagram inspired fromrealistic GoM data. The ANN approximation was found to predictwell the fatigue results, with a maximum discrepancy of 16% onthe predicted fatigue life.

This pilot study has demonstrated that fatigue life calculationsfor the touchdown zone of SCRs may potentially be reduced to amatter of minutes using the proposed ANN framework, withoutcompromising much on the level of accuracy, and without the needfor advanced numerical analyses. Once ‘trained’, the ANN approx-imation may be used conveniently by any external user and couldrepresent a major improvement in efficiency of SCR fatigue estima-tions, particularly for the early stages of design where optimisationstudies are needed to establish values of input parameters thatprovide optimal performance.

Acknowledgements

This work forms part of the activities of the Centre for OffshoreFoundation Systems (COFS), currently supported as a node of theAustralian Research Council Centre of Excellence for GeotechnicalScience and Engineering and as a Centre of Excellence by the

Lloyd’s Register Foundation (LRF). Lloyd’s Register Foundationhelps to protect life and property by supporting engineering-re-lated education, public engagement and the application ofresearch. The first author acknowledges her research studentshipsupport from The University of Western Australia (UWA) and thesupport she received from the members of the Centre for AppliedStatistics at UWA through the postgraduate clinic activity. Theauthors would also like to thank W/Prof. Mike Efthymiou fromthe Shell EMI Chair of Offshore Structures, UWA, and ShellHouston and Shell Rijswijk for their feedback on the study.

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