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Sensitivity studies of SCR fatigue damage in the touchdown zone using an efficient simplified framework for stress range evaluation
Ocean Engineering 96 (2015) 295–311
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Sensitivity studies of SCR fatigue damage in the touchdown zone using an efficient simplified framework for stress range evaluation Lucile M. Quéau n, Mehrdad Kimiaei, Mark F. Randolph Centre for Offshore Foundation Systems, The University of Western Australia, Crawley, WA 6009, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 14 July 2014 Accepted 20 December 2014 Available online 30 January 2015
Steel catenary risers (SCRs) are widely used in deep water. Several sources of nonlinearities make SCR fatigue design challenging. Limited understanding of the influence of the various input parameters on the structural response of SCRs leads to unnecessarily high conservatism in design. Also, time consuming numerical simulations are usually performed to assess SCR fatigue damage which is inefficient, especially for early design stages. A simplified framework for fatigue analysis of SCRs in the touchdown zone (TDZ) has been developed previously, using artificial neural networks. The approach may be used to efficiently estimate maximum static and dynamic stress ranges in the TDZ, from which the fatigue damage can be deduced. Comparison of the maximum static and dynamic stress changes for a given input motion allows quantification of the dynamic amplification factor (DAF). This paper explores the sensitivity of the maximum dynamic stress ranges and DAF to the key dimensionless groups of input parameters and also certain individual input parameters. The study illustrates the usefulness of the proposed framework in understanding SCR behaviour in the TDZ, providing guidance on optimisation of SCR design from a fatigue perspective. The paper also reflects on the potential benefits of using DAFs for SCR fatigue design. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Steel catenary risers Fatigue Sensitivity analyses Artificial neural network Dynamic amplification factor
1. Introduction Oil and gas developments in deep water commonly use steel catenary risers (SCRs) to transfer fluids between seabed and sea surface as they are a cost effective solution. Under the action of environmental loading a complex interaction is created in the touchdown zone (TDZ—the dynamic area of riser-soil interaction (Bridge, 2005)) and fatigue damage is generated. Fatigue design is a major challenge for SCRs as there is a lack of understanding of the influence of the various input parameters on the fatigue damage, leading to unnecessary conservatism. Fatigue damage is usually evaluated by running time consuming numerical simulations. This approach is inefficient, especially for the early stages of design where optimisation studies are required to find values of input parameters leading to the best performance. A simple approach able to provide quantitative guidance on how the input parameters impact the static and dynamic response of SCRs in the TDZ, thus allowing rapid evaluation of the fatigue damage in the TDZ, would be better suited for initial design. The development of such an approach was indeed encouraged in a recent standard for the design of offshore risers (DNV-OS-F201, 2010).
n
Corresponding author. Fax: þ 61 8 6488 1044. E-mail address: [email protected] (L.M. Quéau).
http://dx.doi.org/10.1016/j.oceaneng.2014.12.038 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
A programme of research (Quéau, 2014; Quéau et al., 2011, 2013, 2014a, 2014b, in preparation) has been undertaken by the authors with the ultimate aim of proposing a simplified fatigue analysis and design framework for SCRs in the TDZ. The original intention was to use the dynamic amplification factor (DAF) approach for dynamic response of SCRs, since it is used widely for simplification of structural dynamic analyses for linear structural systems (e.g. Barltrop and Adams, 1991; Bea et al., 1999; Ruiz-Teran and Aparicio, 2006). DAFs quantify the amplification of stress due to dynamic effects when compared with the static response. The simplification relies on the ability to evaluate the static response and the DAF values through simple methods, and hence deduce the dynamic response. However, SCR behaviour is impacted by geometrical and material nonlinearities in the TDZ and therefore the usefulness of the DAF approach for SCR design was explored first. The study is limited to 2D conditions where the current profile in the sea column, the rotational stiffness at the hang-off point (HOP), the coating and the flow rate of the content are not taken into account. Only the case of a linear soil model, horizontal and flat seabed is investigated here. Also, an illustration of the main input parameters is given in Fig. 1 and the following parameters remain unchanged in the study:
Gravity acceleration (g¼ 9.81 m/s2), Young's modulus and Poisson's ratio for the SCR (E ¼2.12E8 kPa, ν ¼ 0.293), water and steel density (ρwater ¼ 1.025 te/m3; ρsteel ¼7.85 te/m3),
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soil friction coefficient (μ ¼0.5), hydrodynamic coefficients (CD ¼1.2; CA ¼ 1), angular position on the SCR circumference (β ¼01), angle of the motion relative to the hang-off angle (Δθm ¼ θm θHO ¼01, as detailed hereafter).
The loading of SCRs is represented by harmonic motions of the floating vessel through the procedure suggested by Kimiaei et al. (2010), where a time varying sinusoidal motion of the vessel, making an angle θm with the vertical and characterised by heave amplitude (H) and period (T) of the motion (i.e. maximum input velocity of 2πH/T) is applied. The vessel is always a semisubmersible
in this study, with its motion applied parallel to the riser tangential line at the hang-off point (hence Δθm set to zero). This is based on the findings from Kimiaei et al. (2010) that, with a semisubmersible, the tangential component of motion alone could adequately represent the fatigue damage resulting from wave loading in the TDZ. Under these assumptions, a pilot study was performed, defining the DAF for SCRs as: DAF¼Max ΔσTDZ_dyn/Max ΔσTDZ_sta, where Max ΔσTDZ_sta and Max ΔσTDZ_dyn are the maximum stress ranges occurring in the TDZ under static and dynamic loading respectively (Quéau et al., 2011). Since the study indicated that the DAF approach was well suited to SCR fatigue design in the TDZ, the more substantial steps necessary to develop the approach were pursued. The major challenge was to predict DAF values and their sensitivity to key input parameters. As a pre-requisite to the sensitivity studies, dimensional analysis was carried out for SCR systems under harmonic motions to identify the dimensionless groups of input parameters influencing the stress range, and consequently the DAF (Quéau et al., 2013). A summary of the various dimensionless groups is given in Fig. 2, using the dimensionless groups notation introduced in Quéau et al. (2013). The validity of the dimensionless groups was tested through numerical analyses and demonstrating that, under the simplifying assumptions adopted in the study, the maximum stress ranges occurring respectively under dynamic motions (accounting for hydrodynamic and inertia effects as longitudinal and transverse waves travel along the riser) and static motions (no inertia or damping effects) could be expressed as follows: Max Δσ TDZ_dyn H ¼f ; Δθm ; E Δz
Fig. 1. SCR geometry and key parameters: (a) side view of SCR configuration; and (b) Riser cross-section (Quéau et al., 2013).
H T
rffiffiffiffiffiffiffiffiffiffi
ρsteel Do Do p To ks ρ g Δzρsteel ; ;β ; ; ; ν; ; μ; ; C D ; C A ; steel ; E Δz w t E Δz E Δz 2 E ρwater E
ð1Þ
Fig. 2. Flowchart of the ANN framework.
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Table 1 Selected extreme values of the individual input parameters. Varied input parameter
Minimum value
Maximum value
Water depth, Δz Outside diameter, Do Wall thickness, wt
400 m 0.1524 m Do/15 if 0.1524 mr Do o 0.36 m Do/20 if 0.36 m rDo o0.56 m Do/25 if 0.56 m rDo r0.762 m 0 kg/m3
2000 m 0.762 m Do/9 if 0.1524 m r Do r 0.36 m Do/12 if 0.36 mo Do r 0.56 m Do/15 if 0.56 m oDo r0.762 m 1025 kg/m3 if 0.1524 mr Do r 0.46 m 800 kg/m3 if 0.46 mo Do r 0.762 m Such that: θHO ¼201 if 400 mr Δz r950 m θHO ¼171 if 950 m oΔz r 1500 m θHO ¼111 if 1500 m o Δzr 2000 m
Content density, ρcont Horizontal tension component, To
Such that: θHO ¼ 171 if 400 mr Δzo 950 m θHO ¼ 91 if 950 mr Δz o 1500 m θHO ¼ 71 if 1500 m rΔz r 2000 m
Heave amplitude, H Soil stiffness, ks Period of the input motion, T
0.1 m 11.4 kPa 4 s if 0.1 mr Ho 1 m 7 s if 1 mr Ho 3.5 m 10 s if 3.5 m rH o5.5 m 13 s if 5.5 m rH r7.5 m
7.5 m 228 kPa 20 s
Table 2 Selected ranges of the dimensionless groups for SCR stress analysis. Varied dimensionless group
Minimum value ( )
Maximum value ( )
Riser displacement amplitude, π2 ¼ H/Δz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Riser displacement velocity, π4 ¼ðH=TÞ ρsteel =E Riser outside diameter, π5 ¼ Do/Δz Riser outside diameter to wall thickness ratio, π6 ¼ Do/wt Riser unit submerged weight, π7 ¼ p/(EΔz) Riser tension, π9 ¼ To/(EΔz2) Soil stiffness, π11 ¼ ks/E Water depth, π15 ¼ gΔzρsteel/E
5.00E 05 9.62E 07
1.88E 02 1.11E 4
7.62E 05 9 3.92E 13 5.44E 14 5.38E 08 1.45E 04
1.91E 03 25 8.07E 11 4.19E 11 1.08E 06 7.26E 04
Table 3 Parts removed of the design space for dynamic loading. Range of Δz (restricting the range of π15)
Area excluded from the design space for dynamic loading (for any appropriate values of π4, π5, π6, π7, π9, π11)
400 m r Δzr 950 m 950 mo Δz r1500 m 1500 m oΔz r 2000 m
π2 r 6.88E 4 π2 r 1.89E 4 π2 r 1.27E 4
Max Δσ TDZ_sta H ¼f ; Δθ m ; E Δz Do Do p To ks ρ gΔzρsteel ; ; ; ν; ;β ; μ; ; steel ; Δz wt EΔz EΔz2 E ρwater E
ð2Þ
β is the angular position on the SCR circumference, Δz is the Δθm is the angle of the motion relative to the hang-off angle (θHO), μ is the soil friction coefficient, ν is Poisson's ratio, ρsteel and ρwater are the steel and
where
vertical difference between hang-off point and seabed,
water densities, CD and CA are the drag and added mass coefficients, Do are wt, are the riser outer diameter and wall thickness, E is Young's modulus, g is the gravity acceleration, H and T are the heave amplitude and period of the input motion, ks is the soil stiffness, p is the unit submerged weight, and To is the horizontal tension component. The dimensionless groups of parameters impacting the value of DAF are the same as those for the maximum dynamic stress range, because of the DAF definition. Throughout the study, numerical analysis of the SCR system has been carried out using the dynamic
analysis software OrcaFlex (Orcina, 2011). Approximate, but accurate, analytical solutions can evaluate the static response (and maximum static stress range) of an SCR (Quéau et al., 2014a). However, since it is complex to develop analytical solutions able to accurately evaluate the dynamic response of SCRs in time domain, the approach has been to develop artificial neural networks (ANNs), trained using the numerical results from OrcaFlex. This approach was validated initially for the static response (Quéau et al., 2014b) before embarking on the dynamic response. Extensive sensitivity studies were performed to capture SCR behaviour in the TDZ for many thousands of different configurations, loading conditions, riser properties etc. (Quéau et al., 2014a, 2014b, Under review). The sensitivity studies were originally intended to increase the understanding of SCR behaviour and predict the DAF value using an advanced mathematical tool, the artificial neural network (ANN). Since the same amount of computational effort was required for approximating the relationships between either Max ΔσTDZ_dyn or DAF, it was decided to use the ANN tool to capture the relationships between the input dimensionless groups and Max ΔσTDZ_dyn directly. Indeed, this is essentially more convenient for the final user in future applications (the maximum dynamic stress range being directly linked to the fatigue damage). The development of such a framework was performed in Quéau et al. (2014a, 2014b, Under review). The main aim of this paper is to link together all of the previous steps of the study, focusing on two distinct aspects: (i) After presenting the ANN framework (Section 2), it is used to efficiently explore the sensitivity of the maximum stress range in the TDZ to variations of some of the input dimensionless
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Table 4 Performance of the approximations from the ANN framework.
Database size Proportion of cases with errors within:
75% 715%
Reference
groups and individual parameters (Section 3); the results are then used to optimise an example SCR system in respect of the fatigue life (Section 4). (ii) Reflecting back on the initial strategy of combining the static response with a dynamic amplification factor, the ANN framework is used to examine the sensitivity of the DAF to some of the input dimensionless groups and individual parameters (Section 3). The key question to resolve is whether the DAF (together with the static response) is more predictable than the maximum dynamic stress range in the TDZ directly, and thus would offer a more robust design approach, just as commonly used for other types of structural design involving dynamic response.
2. The simplified framework 2.1. Selected range of applicability Under the assumptions of the study, there are respectively seven and eight key input dimensionless groups influencing respectively the maximum stress ranges in the TDZ for SCRs under static and dynamic conditions. These dimensionless groups are as follows (using dimensionless groups notation detailed in Fig. 2 (Quéau et al., 2013)):
Riser displacement, π2 ¼ H/Δz. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Riser displacement velocity, π4 ¼ðH=TÞ ρsteel =E (only for
dynamic motion). Riser outside diameter, π5 ¼Do/Δz. Riser outside diameter to wall thickness ratio, Riser unit submerged weight, π7 ¼p/(EΔz). Riser tension, π9 ¼ To/(EΔz2). Soil stiffness, π11 ¼ks/E. Water depth, π15 ¼ g Δzρsteel/E.
π6 ¼ Do/wt.
To develop a simplified, but broadly applicable framework it was necessary to select wide ranges for these key input dimensionless groups, which define the design space, corresponding to wide ranges of the inherent individual input parameters. The selected ranges are presented in Tables 1 and 2. A series of criteria, referred to as “design criteria” as presented in Table 1, were implemented between some of the input parameters and dimensionless groups to select realistic combinations of dimensionless groups when defining SCR configurations and loading conditions. For instance, small hang-off angles were considered for deep water and larger for shallower water to match with industry practices. The riser displacement characteristics were derived from in-service conditions of a typical semisubmersible vessel under calm to harsh seastates in the Gulf of Mexico (GoM). For the dynamic motion, small parts of the design space had to be excluded since it was not possible at this stage to define an approximation of the relationships between the input dimensionless groups and the maximum normalised stress range in the TDZ with a sufficient level of accuracy in these areas, as discussed later. These areas are summarised in Table 3. Further details on the choice of ranges can be found in Quéau et al. (2014a, 2014b, Under review).
9-ANNs static approximation
9-ANNs dynamic approximation
4 50,000 4 99% 100% Quéau et al., 2014b
4 40,000 4 86% 4 97% Quéau et al., Under review
2.2. Framework characteristics A global ANN approximation for the entire design space was developed for the static loading cases initially, in order to ensure the usefulness of the proposed methods, and the more complex dynamic loading cases were examined afterwards. Ultimately, two approximations were established, namely the “9-ANNs static approximation” (Quéau et al., 2014b) and the “9-ANNs dynamic approximation” (Quéau et al., Under review). Both approximations comprise a total of nine single hidden layer Levenberg–Marquardt back-propagation neural networks, having two activation functions (a bipolar sigmoidal function for the nodes of the hidden layer and an identity function for the output nodes) and different number of neurons in the hidden layer (ranging from 20 for some of these ANNs to 100 for others). Each of these ANNs were trained and tested using large databases to find the weights and biases matrices leading to a good match between approximated results and results from OrcaFlex simulations (Quéau et al., 2014b, Under review). These approximations were developed as standalone MATLAB applications and have been combined into a single standalone MATLAB application, namely “ANN framework”, for convenience.1 The flowchart for the application is given in Fig. 2. All the matrices of weights and biases inherent to the various ANNs are built-in the standalone application for straightforward use. The application uses an Excel™ interface so that a user can simply copy and paste a dataset (i.e. series of dimensionless groups' combinations defining SCR configurations and loading conditions, as per the “Varied dimensionless groups” in Fig. 2) and automatically obtain the maximum stress range corresponding to the static and dynamic loading of the specified SCRs and loading conditions (as per the “Approximated outputs” in Fig. 2). The standalone application performs matrix multiplications based on the input values and the built-in matrices of weights and biases to calculate the maximum stress range results (Quéau et al., 2014b, Under review), and therefore the time taken to develop the approximations does not impact the time necessary to assess the outputs with the ANN framework application. The fatigue damage can then be evaluated directly from the dynamic results. The ANN framework application also reports DAF values, evaluated using the static and dynamic results obtained with the 9-ANNs static and dynamic approximations. 2.3. Performance of the ANN framework The ANN framework application is very efficient, taking about 1 min to estimate the outputs (Max ΔσTDZ_sta, Max ΔσTDZ_dyn and DAF) for 50,000 cases (different SCR configurations with given static or dynamic loading conditions) and without the need for any specialised software. It is difficult to quantify accurately the time saving impact of the ANN framework since the time necessary to obtain stress range results from marine analysis software (such as OrcaFlex, as used here) will vary from user to user, depending on the experience of the user, the performance of the computer used to run the simulations and on the efficiency of pre- and postprocessing the numerical models. The last aspect may be 1
The application may be obtained through the corresponding author.
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streamlined by the use of automation sub-routines (Quéau et al., 2014a, 2014b, Under review). However, four months of calculation were necessary here to obtain the results for the entire database using OrcaFlex running on a high performance computer. The proposed simplified framework represents therefore a tremendous time saving in fatigue design studies, with only limited loss in accuracy. The two approximations provide a good level of accuracy as summarised in Table 4. Excellent accuracy was consistently reached with the 9-ANNs static approximation. With the 9ANNs dynamic approximation, a reasonably good level of accuracy was obtained, although future work (outside the scope of this paper) was recommended to improve the accuracy further in order to increase the robustness of the predictions (Quéau et al.,
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Under review). There is also a restriction on minor parts of the design space for the 9-ANNs dynamic approximation where the results might not be as accurate as for the rest of the design space and therefore it is not recommended to apply the approximation in these areas. This is the case for regions corresponding to low values of imposed displacement relative to the water depth (π2), as shown in Table 3; these regions represent waves that are unlikely to contribute significantly to the fatigue damage (Quéau et al., Under review). The performances indicated in Table 4 are valid when excluding the low π2 values, as presented in Table 3, from the design space. Application of the 9-ANNs dynamic approximation to a series of SCR configurations was shown to predict the fatigue life within 7 15% of the numerical results (Quéau et al., Under review).
Table 5 Selected values to investigate the sensitivity of SCR behaviour to displacement amplitude and velocity.
Constant
Varied
Input parameter/dimensionless group
Value
Do (m) wt (m) p (kN/m) Δz (m) θHO (1) ks (kPa) ρcont (kg/m3) To (kN) π5 ¼Do/Δz π6 ¼Do/wt π7 ¼p/(EΔz) π9 ¼To/(EΔz2) π11 ¼ks/E π15 ¼ gΔzρsteel/E H (m)
0.4572 0.0286 1.31 1225 13 119.7 0 466 3.73E 04 16 5.05E 12 1.47E 12 5.65E 07 4.45E 04 5 values per subplot: (a) 0.1; 0.325; 0.55; 0.775; 1 (b) 1 þ ; 1.625; 2.25; 2.875; 3.5 (c) 3.5 þ ; 4; 4.5; 5; 5.5 (d) 5.5 þ ; 6; 6.5; 7; 7.5 20 values per subplot, equally spaced within the following ranges: (a) 4–20 (b) 7–20 (c) 10–20 (d) 13–20 5 values per subplot as per appropriate H value:
T (s) π2 ¼H/Δz
(a) (b) (c) (d) π4 ¼ðH=TÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
0.08E 3; 0.27E 3; 0.44E 3; 0.63E 3; 0.82E 3 0.82E 3 þ ; 1.33E 3; 1.84E 3; 2.35E 3; 2.86E 3 2.86E 3 þ ; 3.27E 3; 3.67E 3; 4.08E 3; 4.49E 3 4.49E 3 þ ; 4.9E 3; 5.31E 3; 5.71E 3; 6.12E 3
20 values per selected H value as per appropriate H and T values, in the following ranges: (a)
– – – – –
9.62E 7 to 4.81E 6 for H¼ 0.1 m 3.13E 6 to 1.56E 5 for H¼ 0.325 m 5.29E 6 to 2.65E 5 for H¼ 0.55 m 7.46E 6 to 3.73E 5 for H¼0.775 m 9.62E 6 to 4.81E 5 for H¼ 1 m
(b)
– – – – –
9.62E 6 þ to 2.75E 5 for H¼ 1 þ m 1.56E 5 to 4.47E 5 for H¼ 1.625 m 2.16E 5 to 6.19E 5 for H¼ 2.25 m 2.77E 5 to 7.90E 5 for H¼ 2.875 m 3.37E 5 to 9.62E 5 for H¼ 3.5 m
(c)
– – – – –
3.37E 5 þ to 6.73E 5 for H¼3.5 þ m 3.85E 5 to 7.70E 5 for H¼ 4 m 4.33E 5 to 8.66E 5 for H¼ 4.5 m 4.81E 5 to 9.62E 5 for H¼ 5 m 5.29E 5 to 1.06E 4 for H¼ 5.5 m
(d)
– – – – –
5.29E 5 þ to 8.14E 5 for H¼ 5.5 þ m 5.77E 5 to 8.88E 5 for H¼6 m 6.25E 5 to 9.62E 5 for H¼ 6.5 m 6.73E 5 to 1.04E 4 for H¼ 7 m 7.22E 5 to 1.11E 4 for H¼7.5 m
The superscriptsþ and – in 1 þ or 1 for instance, indicate a value close to 1 but higher or lower than 1 by a negligible amount. They are used to select values to reflect of the design criteria between H and T.
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In the rest of the paper, this simplified framework is used for convenience on a smaller part of the design space where the 9-ANNs dynamic approximation underwent more substantial validations (Quéau et al., Under review). This corresponds to water depths of 950 mo Δzr1500 m (and therefore 3.45E4o π15 r5.45E 4), riser displacement π2 41.89E 4 (due to the comment on lower accuracy and fatigue relevance for low riser displacement) and corresponding ranges for π4, π5, π6, π7, π9, π11. 3. Sensitivity studies using the ANN framework The simplified framework is now used to shed light on the sensitivity of the maximum stress range in SCRs in the TDZ and of the DAF to the variation of some of the key input dimensionless groups of parameters. The resulting sensitivities are benchmarked against results from the published literature. There are many ways to vary the key input dimensionless groups and individual parameters and this paper illustrates the usefulness of the ANN framework by considering a selection of examples of individual and simultaneous variations of the key input dimensionless groups and individual input parameters. The first two examples, testing the effect of the displacement characteristics and the soil stiffness on the maximum stress range in the TDZ, aim to demonstrate that the ANN framework can reproduce expected and known trends to validate further its accuracy. Building on from these two examples, two further examples derived from published studies performed using the traditional approach (i.e. numerical simulations) are presented to illustrate the usefulness of the ANN framework in validating published trends for new ranges of the inputs. Throughout this section and wherever possible based on (i) the tested variations, (ii) the selected values for the input parameters and dimensionless groups, and (iii) the design criteria among them, the unvaried dimensionless groups and individual parameters are fixed to their appropriate mean value, except for π7 the value of which is determined based on the water depth and the riser unit submerged weight for an empty riser (for convenience). The term ‘appropriate’ implies that the design criteria between individual input parameters and dimensionless groups, as shown in Table 2, are respected and refers to the mean values accounting for the selected range of water depth: 950 m o Δzr1500 m (or 3.45E 4 o π15 r5.45E 4). Moreover, the reported values of the horizontal component of the tension (To) is assessed by using one of the common catenary equations: To ¼ pΔzsin(θHO)/(1 sin (θHO)). 3.1. Effect of the imposed riser displacement amplitude (π2) and velocity (π4) The effect of the imposed riser displacement amplitude (π2) and velocity (π4) on SCR behaviour in the TDZ is investigated first. For this purpose, H and T were varied within the four selected ranges of H and corresponding T values. Table 5 summarises the selected values of the individual input parameters and dimensionless groups that were used for this example. A series of design charts showing the sensitivity of Max ΔσTDZ_dyn/E and DAF to variations of π2 and π4 were established, as illustrated in Figs. 3 and 4 respectively. The range of π4 was normalised in the subplots for convenience, using the extreme maximum (max) and minimum (min) values shown in Table 5 (so that Normalised π4 ¼(2* π4 - max - min)/(max - min)). Results are discussed next, for the maximum stress range in the TDZ first and then for the DAF. 3.1.1. Maximum stress range in the TDZ Increasing the velocity (π4) by keeping the displacement amplitude constant (π2), thereby reducing the period of the imposed
displacement (T), leads to an increase of Max ΔσTDZ_dyn/E. Also, the higher the displacement amplitude, the higher the maximum stress range in the TDZ (i.e. increase of fatigue damage) since it generates higher variations of the curvature in the TDZ. These trends are observed for the four ranges of H and corresponding T values selected and show that no resonance effects were detected for the SCR configuration under study.
3.1.2. DAF The values of maximum stress range found under static loading (Max ΔσTDZ_sta/E) using the simplified framework are shown in Table 6. They were used to calculate the DAF values from Max ΔσTDZ_dyn/E results presented in Fig. 3. Fig. 4 indicates that the DAF increases with increasing velocity of the imposed displacement. Also, it tends to confirm an interesting trend of the DAF sensitivity that was already noted at the pilot study stage of this research (Quéau et al., 2011). For low displacement amplitudes, DAF varies inversely with the displacement amplitude, whereas it increases with increasing displacement amplitude for higher heave amplitude values; Fig. 4b captures the transition between these opposing trends. Interestingly, DAF values do not decrease down to unity in any of the plots presented in Fig. 4, indicating that the SCR under study did not reach its static response for the selected displacements and wave periods. To some extent, this is consistent with the results obtained during the pilot study (Quéau et al., 2011). Even though the selected values for the input parameters and dimensionless groups are not all identical in this paper to those selected for the pilot study, DAF values of unity were only reached in the pilot study when considering imposed displacements with a period (T) greater than 20 s. That limit is beyond the highest limit selected for T here and also when defining the ANN framework (see Table 1 and Quéau et al., 2014a, 2014b, Under review). The similarity of the observed trends when using the ANN framework to those of the pilot study, when DAF sensitivity was investigated using results from numerical simulations, further validates the framework. 3.2. Effect of the soil stiffness (π11) for various imposed displacement characteristics This example focuses on the effect of the soil stiffness (π11) on SCR behaviour in the TDZ for various characteristics of the imposed displacement. The values of the individual input parameters and dimensionless groups used for this purpose are summarised in Table 7 and the design charts are shown in Figs. 5 and 6 for Max ΔσTDZ_dyn/E and DAF respectively. 3.2.1. Maximum stress range in the TDZ The overall trend of the results presented in Fig. 5 is for increasing values of Max ΔσTDZ_dyn/E with increasing soil stiffness, which is consistent with other published work (e.g. Bridge et al., 2004; Quéau et al., 2011). The trend results from higher contact forces and greater curvature variation in the TDZ with increasing soil stiffness. There are minor fluctuations in Fig. 5d where higher soil stiffness locally gives lower values of Max ΔσTDZ_dyn/E for low values of displacement velocity. This may however indicate local inaccuracy of the ANN framework rather than an indication of a different trend for the results in that area. As is true for any simplified approach (DNV-OS-F201, 2010), engineering judgement is required when interpreting results from the proposed framework to differentiate between results indicating a change in SCR behaviour and those affected by slight inaccuracies of the simplified framework.
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3.2.2. DAF DAF increases with increasing velocity of the imposed displacement, as observed for all selected values of soil stiffness and
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displacement amplitude (see Fig. 6). Except for high velocity values (i.e. low period of imposed displacement), DAF is almost insensitive to the soil stiffness. This confirms the trend observed in
Fig. 3. Sensitivity of Max ΔσTDZ_dyn/E to the variation of the loading amplitude (π2) and velocity (π4), accounting for the design criteria between H and T: (a) 0.1 m rH o1 m; (b) 1 mr Ho 3.5 m; (c) 3.5 m r Ho5.5 m; and (d) 5.5 mr Hr 7.5 m.
Fig. 4. Sensitivity of DAF to the variation of the loading amplitude (π2) and velocity (π4), accounting for the design criteria between H and T: (a) 0.1 m rHo 1 m; (b) 1 mr Ho 3.5 m; (c) 3.5 m r Ho5.5 m; and (d) 5.5 mr Hr 7.5 m.
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Quéau et al. (2011) with other values of input parameters and dimensionless groups (e.g. different values of water depth, riser diameter and wall thickness, hang-off angle etc…).
Table 6 Static stress range results necessary to explore DAF sensitivity to displacement amplitude and velocity. H (m)
π2 ¼ H/Δz
Max ΔσTDZ_sta/E
0.100 0.325 0.550 0.775 1.000 1.625 2.250 2.875 3.500 4.000 4.500 5.000 5.500 6.000 6.500 7.000 7.500
8.16E 05 2.65E 04 4.49E 04 6.33E 04 8.16E 04 1.33E 03 1.84E 03 2.35E 03 2.86E 03 3.27E 03 3.67E 03 4.08E 03 4.49E 03 4.90E 03 5.31E 03 5.71E 03 6.12E 03
9.80E 06 3.17E 05 5.33E 05 7.46E 05 9.57E 05 1.51E 04 2.02E 04 2.48E 04 2.90E 04 3.21E 04 3.48E 04 3.74E 04 3.97E 04 4.19E 04 4.38E 04 4.56E 04 4.72E 04
3.3. Effect of the water depth (π15) for various hang-off angles and imposed displacement characteristics In this example the effect of increasing water depth (π15) for a constant value of hang-off angle is investigated. Various values of hang-off angle and imposed displacement characteristics are considered. This example was selected to compare the results found with the simplified framework and for the selected ranges of parameters against the results reported by Zhan (2010), established using numerical simulations. The selected values of input parameters and dimensionless groups are presented in Table 8 and the results are shown in Figs. 7–9. The hang-off angle is measured from the vertical.
3.3.1. Maximum stress range in the TDZ The static results are presented in Fig. 7 and indicate that increasing water depth leads to a reduction of Max ΔσTDZ_sta/E. Also, higher hang-off angle values result in lower values of Max ΔσTDZ_sta/E due to increased tension in the TDZ (π9), thereby reducing the curvature variation in the TDZ. The marked change in gradient observed between the two lowest values of water depth (particularly for low hang-off angles) may indicate slight inaccuracy of the ANN approximation rather than reflecting true SCR behaviour. Indeed, estimation of the maximum stress range in
Table 7 Selected values to investigate the sensitivity of SCR behaviour to the variation of the soil stiffness for various displacement characteristics.
Constant
Varied
Input parameter/dimensionless group
Value
ρcont (kg/m3) Do (m) wt (m) p (kN/m) Δz (m) θHO (1) To (kN) π5 ¼ Do/Δz π6 ¼ Do/wt π7 ¼ p/(EΔz) π9 ¼ To/(EΔz2) π15 ¼ gΔzρsteel/E ks (kPa) H (m)
0 0.4572 0.0286 1.31 1225 13 466 3.73E 04 16 5.05E 12 1.47E 12 4.45E 04 5 values in each subplot: 11,4; 65.55; 119.7; 173.85; 228 1 value per subplot: (a) 0.55 (b) 2.25 (c) 4.5 (d) 6.5
T (s)
20 values per subplot, equally spaced within the following ranges: (a) 4–20 (b) 7–20 (c) 10–20 (d) 13–20
π2 ¼ H/Δz
1 value per subplot as per appropriate H value: (a) 4.49E 4 (b) 1.84E 3 (c) 3.67E 3 (d) 5.31E 3
π4 ¼ ðH=TÞ
π11 ¼ ks/E
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
20 values per subplot, equally spaced within the following ranges: (a) 5.29E 6 to 2.65E 5 (b) 2.16E 5 to 6.19E 5 (c) 4.33E 5 to 8.66E 5 (d) 6.25E 5 to 9.62E 5 5 values in each subplot: 5.38E 8; 3.09E 7; 5.65E 7; 8.20E 7; 1.08E 6
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Fig. 5. Sensitivity of Max ΔσTDZ_dyn/E to the variation of the displacement velocity (π4) for various soil stiffnesses (π11) and displacement amplitudes: (a) π2 ¼0.44E 3; (b) π2 ¼ 1.84E 3; (c) π2 ¼ 3.67E 3; and (d) π2 ¼5.31E 3.
Fig. 6. Sensitivity of DAF to the variation of the displacement velocity (π4) for various soil stiffnesses (π11) and displacement amplitudes: (a) π2 ¼0.44E 3; (b) π2 ¼1.84E 3; (c) π2 ¼3.67E 3; and (d) π2 ¼ 5.31E 3.
the TDZ for the extreme values of inputs may not be as accurately captured as for intermediate input values, which is inherent to the response surface method.
For dynamic results, Fig. 8 shows that increasing water depth generally results in a slight reduction of the dynamic stress range (Fig. 8a–c) although an opposite trend is found in Fig. 8d for the
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Table 8 Selected values to investigate the sensitivity of SCR behaviour to the variation of water depth for various hang-off angles and displacement characteristics.
Constant
Varied
Input parameter/dimensionless group
Value
ks (kPa) ρcont (kg/m3) Do (m) wt (m) p (kN/m) π6 ¼Do/wt π11 ¼ks/E Δz (m) θHO (1) H (m)
119.7 0 0.4572 0.0286 1.31 16 5.65E 07 20 values equally spaced within the following range: 950–1500 5 values in each subplot: 9; 11; 13; 15; 17 1 value per subplot: (a) (b) (c) (d)
T (s)
1 value per subplot: (a) (b) (c) (d)
12 13.5 15 16.5
varying as per selected θHO and Δz values, within the following overall range: 231–813 20 values per selected H value, equally spaced within the following ranges: (a) 3.67E 4 to 5.79E 4 (b) 1.50E 3 to 2.37E 3 (c) 3.00E 3 to 4.74E 3 (d) 4.33E 3 to 6.84E 3
To (kN) π2 ¼H/Δz
π4 ¼ðH=TÞ
0.55 2.25 4.5 6.5
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
π5 ¼Do/Δz π7 ¼p/(EΔz) π9 ¼To/(EΔz2) π15 ¼ gΔzρsteel/E
1 value per subplot: (a) 8.82E 6 (b) 3.21E 5 (c) 5.77E 5 (d) 7.58E 5 20 values depending on the value of Δz and equally spaced within the 20 values depending on the value of Δz and equally spaced within the varying as per selected θHO and Δz values, within the following overall 20 values depending on the value of Δz and equally spaced within the
highest displacement amplitude, H¼ 6.5 m. Also, lower hang-off angles lead to higher values of Max ΔσTDZ_dyn/E for all the selected cases expect for H ¼6.5 m where the opposite trend is observed. The trend illustrated in Fig. 8d is consistent with the results from Xia et al. (2008) who found that, for a given water depth, increasing the hang-off angle increased the fatigue damage. Further work is needed to ensure that the abrupt variations exhibited by the results in Fig. 8 reflect true physical behaviour of SCRs in the TDZ and are not the result of the ANN framework. For both static and dynamic conditions, the variation of maximum stress range in the TDZ with water depth is quite limited, showing a rather weak sensitivity for a given hang-off angle. These results are consistent with the results from Zhan (2010), who worked with bending moment (which dominates the contribution to SCR stresses in the TDZ (Shiri and Hashemi, 2012)) and found that the dynamic bending moment envelope did not vary much with changing water depth.
3.3.2. DAF The DAF results are shown in Fig. 9. In general, the DAF increases with increasing water depth and smaller hang-off angles generate less dynamic amplification (Fig. 9a, c and d); a slight opposite trend is found in Fig. 9b, although with low sensitivity of the DAF to the hang-off angle.
following range: 3.05E 4 to 4.81E 4 following range: 8.75E 4 to 1.38E 3 range: 7.65E 13 to 2.69E 12 following range: 3.45E 4 to 5.45E 4
3.4. Effect of the outside diameter (Do) for various wall thicknesses and imposed displacement characteristics This example was inspired by a parametric study performed by Xia et al. (2008) where a SCR base case was established to test the effect of the internal diameter on SCR behaviour in the TDZ. For this purpose, the same loading was applied to the SCR base case for fixed values of hang-off angle (θHO) and wall thickness (wt) but various values of the internal diameter. For the constant wall thickness this resulted in varying the outside diameter (Do), the submerged weight (p) and the horizontal tension component (To) to keep the hang-off angle unchanged. Here, the sensitivity of SCR behaviour to variations of the outside diameter is investigated by considering several values of the wall thickness and also for various displacement characteristics (and corresponding values of p and To), therefore exploring wider ranges of the input parameters. The values of dimensionless groups and individual input parameters used for this purpose are shown in Table 9. Results are presented in Figs. 10–12 and are discussed below.
3.4.1. Maximum stress range in the TDZ In order to compare the results against the findings from Xia et al. (2008), the effect of the variation of the outside diameter on the maximum stress range occurring in the TDZ is illustrated for static and dynamic loading in Figs. 10 and 11 respectively.
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Fig. 7. Sensitivity of Max ΔσTDZ_sta/E to the variation of the water depth (π15) for various hang-off angles (θHO) and displacement characteristics: (a) H¼ 0.55 m; (b) H¼2.25 m; (c) H ¼4.50 m; and (d) H¼ 6.50 m.
Fig. 8. Sensitivity of Max ΔσTDZ_dyn/E to the variation of the water depth (π15) for various hang-off angles (θHO) and displacement characteristics: (a) H¼ 0.55 m and T ¼12 s; (b) H¼2.25 m and T ¼ 13.5 s; (c) H¼4.50 m and T ¼ 15 s; and (d) H¼6.50 m and T ¼ 16.5 s.
In general, increasing the outside diameter for a given wall thickness tends to increase Max ΔσTDZ_sta/E, which is consistent with the results from Xia et al. (2008). However, it seems that for low values of the displacement amplitude a slight opposite trend
may be found (Fig. 10a). Greater wall thickness systematically leads to higher Max ΔσTDZ_sta/E in this example since it results in higher value of submerged weight (π7) and therefore an increase of curvature in the TDZ.
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Fig. 9. Sensitivity of DAF to the variation of the water depth (π15) for various hang-off angles (θHO) and displacement characteristics: (a) H¼0.55 m and T ¼12 s; (b) H¼ 2.25 m and T ¼ 13.5 s; (c) H¼ 4.50 m and T ¼ 15 s; and (d) H¼ 6.50 m and T¼ 16.5 s.
Regarding dynamic loading, Max ΔσTDZ_dyn/E also increases with increasing outside diameter for most of the selected cases, although it decreases for high displacement amplitude and low wall thickness (Fig. 11d). In their case study, Xia et al. (2008) found that the fatigue damage for dynamic loading decreased with increasing outside diameter for a given wall thickness. Further work is therefore needed to validate the observed trends and find the values of input parameters that lead to a change of trends. Another complex behaviour is also noted depending on the value of the displacement amplitude: for low displacement amplitude, the higher the wall thickness, the greater the fatigue damage (Fig. 11a and b), whereas the reverse is true for higher values of displacement amplitude (Fig. 11c and d). This illustrates the usefulness of the simplified framework for the early stages of design where designers can optimise values of wall thickness and outside diameter for the waves that contribute most to the overall fatigue damage under the given wave scatter diagram.
ranges of input parameters and dimensionless groups. The examples have illustrated that conclusions obtained from a base case or for small ranges of input parameters and dimensionless groups cannot always be generalised, complicating general quantification of the influence of particular dimensionless groups on the maximum stress range in the TDZ. This reinforces the need for a simplified method that can conveniently assist screening tasks in the early design stages, providing designers with a tool to optimise values of the dimensionless groups within their control (e.g. riser properties) for given values of dimensionless groups that are outside their control (e.g. environmental conditions). In addition, for the selected examples it was shown that DAF values could vary up to values exceeding 16. Overall, the DAF patterns found in the design charts were no simpler than the patterns for the maximum stress range values. This supports the choice to develop an ANN framework to predict Max ΔσTDZ_dyn/E directly rather than to predict DAF values (Quéau et al., Under review).
3.4.2. DAF Based on the maximum stress ranges in the TDZ, DAF sensitivity to the variation of outside diameter depends on the imposed displacement characteristics. For low displacement amplitude, DAF increases with increasing outside diameter and is higher for greater wall thickness values (Fig. 12a and b). However, for higher displacement amplitudes, DAF decreases with increasing outside diameter and lower wall thicknesses lead to higher DAF values (Fig. 12c and d). Large DAF values in excess of 16 are observed for the highest selected displacement amplitude, the lowest selected wall thickness and the lowest selected outside diameter. The four examples selected in this section were used to compare the trends observed when using the proposed simplified framework against published results obtained with different methods, validating further the usefulness of the framework. It also enabled the sensitivity of SCR behaviour in the TDZ to be explored for wider
4. Optimisation of the fatigue life using the ANN framework This section illustrates application of the ANN framework for SCR fatigue design by performing a simple screening study typical of initial design. The case study is based on deterministic fatigue design of an inservice SCR in the GoM, namely ‘Base Case 1’ (BC1), characteristics of which are presented in Table 10. A simplified wave scatter table was used and 15 load cases (LCs) characterised by a sinusoidal vessel motion (displacement of amplitude H and period T) were selected to represent the loading conditions. These LCs were derived from a selection of 15 waves from a sample wave scatter table for GoM, with wave heights, periods and number of occurrences presented in Table 11, which represent over 95% of the waves occurring over a 20 year period. The values of the corresponding heave amplitudes were assessed based on RAO
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Table 9 Selected values to investigate the sensitivity of SCR behaviour to the variation of outside diameter for various wall thicknesses and displacement characteristics. Input parameter/dimensionless group Constant Δz (m) θHO (1) ks (kPa) ρcont (kg/m3) π11 ¼ ks/E π15 ¼ gΔzρsteel/E Varied Do (m)
Value
1225 13 119.7 0 5.65E 07 4.45E 04 20 values per selected wt value, equally spaced within the following ranges (selected to respect design criteria between value of Do and π6): – – – – –
wt (m) H (m)
0.152 to 0.24 for wt ¼ 0.016 m 0.162 to 0.27 for wt ¼ 0.018 m 0.18 to 0.3 for wt ¼ 0.02 m 0.198 to 0.33 for wt ¼ 0.022 m 0.216 to 0.36 for wt ¼ 0.024 m
5 values in each subplot: 0.016; 0.018; 0.02; 0.022; 0.024 1 value per subplot: (a) (b) (c) (d)
T (s)
1 value per subplot: (a) (b) (c) (d)
To (kN)
121 to 146 for wt ¼0.016 m 149 to 185 for wt ¼ 0.018 m 184 to 229 for wt ¼ 0.02 m 223 to 277 for wt ¼ 0.022 m 265 to 330 for wt ¼0.024 m
20 values per selected wt value, equally spaced within the following ranges:
p (kN/m)
– – – – – π2 ¼ H/Δz
0.34 0.42 0.52 0.64 0.75
to to to to to
0.41 for wt ¼ 0.016 m 0.52 for wt ¼0.018 m 0.64 for wt ¼0.02 m 0.78 for wt ¼0.022 m 0.93 for wt ¼0.024 m
1 value per subplot as per appropriate H value: (a) (b) (c) (d) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
π5 ¼ Do/Δz
4.49E 4 1.84E 3 3.67E 3 5.31E 3
1 value per subplot as per appropriate H and T value: (a) (b) (c) (d)
8.82E 6 3.21E 5 5.77E 5 7.58E 5
20 values per selected wt value, equally spaced within the following ranges: – – – – –
π6 ¼ Do/wt
12 13.5 15 16.5
20 values per selected wt value, equally spaced within the following ranges: – – – – –
π4 ¼ ðH=TÞ
0.55 2.25 4.5 6.5
1.23E 4 1.32E 4 1.47E 4 1.62E 4 1.76E 4
to 1.96E 4 for wt ¼ 0.016 m to 2.20E 4 for wt ¼ 0.018 m to 2.45E 4 for wt ¼ 0.02 m to 2.69E 4 for wt ¼ 0.022 m to 2.94E 4 for wt ¼ 0.024 m
20 values per selected wt value, equally spaced within the following ranges:
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Table 9 (continued ) Input parameter/dimensionless group
Value
– – – – – π7 ¼ p/(EΔz)
20 values per selected wt value, equally spaced within the following ranges: – – – – –
π9 ¼ To/(EΔz2)
9.4 to 15 for wt ¼ 0.016 m 9 to 15 for wt ¼ 0.018 m 9 to 15 for wt ¼ 0.02 m 9 to 15 for wt ¼ 0.022 m 9 to 15 for wt ¼ 0.024 m
1.31E 12 to 1.59E 12 for wt ¼ 0.016 m 1.62E 12 to 2.01E 12 for wt ¼ 0.018 m 2.00E 12 to 2.48E 12 for wt ¼0.02 m 2.41E 12 to 3.00E 12 for wt ¼ 0.022 m 2.87E 12 to 3.57E 12 for wt ¼ 0.024 m
20 values per selected wt value, equally spaced within the following ranges: – – – – –
3.82E 13 to 4.60E 13 for wt ¼ 0.016 m 4.69E 13 to 5.83E 13 for wt ¼ 0.018 m 5.79E 13 to 7.20E 13 for wt ¼ 0.02 m 7.01E 13 to 8.71E 13 for wt ¼0.022 m 8.34E 13 to 1.04E 12 for wt ¼0.024 m
Fig. 10. Sensitivity of Max ΔσTDZ_sta/E to the variation of outside diameter (Do) for various wall thicknesses (wt) and displacement characteristics: (a) H¼ 0.55 m; (b) H¼ 2.25 m; (c) H¼ 4.50 m; and (d) H¼ 6.50 m.
(response amplitude operator) tables of the semisubmersible following the procedure proposed by Kimiaei et al. (2010). It is assumed that a designer wishes to optimise the outside diameter (Do), outside diameter over wall thickness ratio (π6) and hang-off angle (θHO) for the riser presented in BC1 in order to improve the fatigue life. The location of the SCR is fixed so that the loading conditions (displacement amplitude and velocity, π2 and π4
respectively), the soil stiffness (π11) and the water depth (π15) are the same as for BC1 and also the content density (ρcont) is fixed. The only parameters in the designers' control are therefore the following dimensionless groups: π5, π6 and π9 (π7 varying accordingly). The example sensitivity studies in the previous section demonstrated that loading conditions might lead to different choices of recommended values of θHO, Do and wt in order to decrease the
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Fig. 11. Sensitivity of Max ΔσTDZ_dyn/E to the variation of outside diameter (Do) for various wall thicknesses (wt) and displacement characteristics: (a) H¼ 0.55 m and T ¼12 s; (b) H¼2.25 m and T ¼ 13.5 s; (c) H¼4.50 m and T ¼ 15 s; and (d) H¼6.50 m and T ¼ 16.5 s.
Fig. 12. Sensitivity of DAF to the variation of outside diameter (Do) for various wall thicknesses (wt) and displacement characteristics: (a) H¼ 0.55 m and T ¼12 s; (b) H¼ 2.25 m and T ¼13.5 s; (c) H¼ 4.50 m and T ¼15 s; and (d) H¼ 6.50 m and T ¼16.5 s.
maximum dynamic stress range in the TDZ. The fatigue life is based on the cumulative damage from each LC of a wave scatter diagram, and the damage is non-linearly related to the maximum dynamic stress
range in the TDZ through the allowable number of occurrence. It is therefore necessary to find the combination of θHO, Do and π6 values leading to the least overall damage to find the optimum fatigue life.
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For the sake of simplicity, it is assumed that the designer has the choice between three values for each of the inputs, as follows:
Table 10 SCR base cases characteristics. Input parameter/dimensionless group
BC1
Do (m) wt (m) p (kN/m) Δz (m) θHO (1) ks (kPa) ρcont (kg/m3) To (kN) π5 ¼ Do/Δz π6 ¼ Do/wt π7 ¼ p/(EΔz) π9 ¼ To/(EΔz2) π11 ¼ ks/E π15 ¼ gΔzρsteel/E
0.228 ( 9 in.) 0.025 ( 1 in.) 0.82 982 9.8 22.8 0 164 2.32E 04 9.12 3.92E 12 8.05E 13 1.08E 07 3.57E 04
Table 11 Loading conditions for the case studies. Period Number of π2 ¼H/Δz Load Wave Heave occurrence Case height Amplitude T (s) for 20 years H (m) (LC) (m) LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10 LC11 LC12 LC13 LC14 LC15 a
1 1 1 1 3 3 3 8 13 8 8 13 18 13 18
0.004 0.15 0.25 0.39 0.44 0.76 1.17 1.17 1.91 2.02 3.12 3.29 4.55 5.06 7.01
3 8 13 18 8 13 18 8 8 13 18 13 13 18 18
156,673,082 10,946,661 1,132,827 305,883 2,795,859 130,073 16,564 68,228 2073 7409 607 647 92 47 4
4.39E 06a 1.49E 04a 2.58E 04 3.97E 04 4.48E 04 7.73E 04 1.19E 03 1.19E 03 1.94E 03 2.06E 03 3.17E 03 3.35E 03 4.64E 03 5.16E 03 7.14E 03
π4 ¼ ðH=TÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
2.76E 07 3.53E 06 3.75E 06 4.16E 06 1.06E 05 1.12E 05 1.25E 05 2.82E 05 4.58E 05 3.00E 05 3.33E 05 4.87E 05 6.74E 05 5.41E 05 7.50E 05
Value of π2 outside the selected range of applicability of the ANN framework.
θHO can take a value of 9.81 (as for BC1), 131 or 171. Do can take a value of 0.228 m (as for BC1), 0.25 m or 0.35 m. π6 can take a value of 9.12 (as for BC1), 12 or 15. Since there are three possible choices for each of three inputs, 27 possible designs result for the SCR. The ANN framework is applied to investigate the fatigue lives of the SCRs formed by the 27 combinations of inputs, with results shown in Table 12. Fatigue results are assessed using the DNV S–N curve type D for seawater (DNV-RP-C203, 2011). Under those circumstances, by keeping Do and π6 to their values in BC1 and increasing θHO to 171 the fatigue life can be increased from 163 to 722 years, improving fatigue performance by a factor greater than 4. This illustrates how the proposed ANN framework can assist screening design tasks in finding combinations of input parameters that optimise fatigue performance, although other practical considerations may affect the final selection of input parameters. In practice, the ANN framework may be used in conjunction with optimisation software in order to consider many more choices for the input parameters.
5. Conclusions This paper has illustrated the usefulness of the simplified framework based on artificial neural networks (ANNs) developed previously (Quéau et al., 2014b, Under review) for SCR design with respect to fatigue in the TDZ. A series of sensitivity studies were performed by varying a selection of dimensionless groups and individual input parameters to illustrate the trends in SCR behaviour. Estimated trends were benchmarked against published results to validate further the accuracy of the ANN framework. The sensitivity studies explored wider ranges of the inputs and showed how misinterpretations can arise from generalising the trends obtained from more restricted studies, since
Table 12 Screening of SCR characteristics for fatigue performance. Combination
Do (m)
π6 ¼ Do/wt
θHO (1)
π5 ¼ Do/Δz
π7 ¼ p/(EΔz)
π9 ¼To/(EΔz2)
p (kN/m)
To (kN)
Fatigue life (yr)
1 (BC1) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35
9.12 9.12 9.12 12 12 12 15 15 15 9.12 9.12 9.12 12 12 12 15 15 15 9.12 9.12 9.12 12 12 12 15 15 15
9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 13 13 13 13 13 13 13 13 13 17 17 17 17 17 17 17 17 17
2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04
3.92E 12 4.72E 12 9.25E 12 2.64E 12 3.18E 12 6.23E 12 1.79E 12 2.15E 12 4.21E 12 3.92E 12 4.72E 12 9.25E 12 2.64E 12 3.18E 12 6.23E 12 1.79E 12 2.15E 12 4.21E 12 3.92E 12 4.72E 12 9.25E 12 2.64E 12 3.18E 12 6.23E 12 1.79E 12 2.15E 12 4.21E 12
8.05E 13 9.68E 13 1.90E 12 5.42E 13 6.52E 13 1.28E 12 3.66E 13 4.41E 13 8.64E 13 1.14E 12 1.37E 12 2.68E 12 7.67E 13 9.22E 13 1.81E 12 5.19E 13 6.23E 13 1.22E 12 1.62E 12 1.95E 12 3.82E 12 1.09E 12 1.31E 12 2.57E 12 7.38E 13 8.87E 13 1.74E 12
0.82 0.98 1.93 0.55 0.66 1.30 0.37 0.45 0.88 0.82 0.98 1.93 0.55 0.66 1.30 0.37 0.45 0.88 0.82 0.98 1.93 0.55 0.66 1.30 0.37 0.45 0.88
164 198 388 111 133 261 75 90 176 233 280 549 157 188 369 106 127 250 331 398 781 223 268 526 151 181 356
163 169 207 134 129 146 113 107 111 313 322 384 216 210 229 157 150 148 722 708 645 400 403 393 226 233 221
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the effect of certain inputs on the maximum stress range in the TDZ may depend on the value of other inputs. For this reason, and because the fatigue life depends on the cumulative damage for a given wave scatter diagram, optimisation of an SCR design with respect to fatigue must consider the overall level of damage rather than individual load cases. The usefulness of the ANN framework for such optimisation was illustrated by a simple example. The sensitivity of DAF values, quantifying the dynamic amplification compared with static results, was also investigated. It was shown that values of DAF as high as 16 could occur with the selected examples and that the DAF pattern was no easier to predict than the pattern of the maximum dynamic stress range in the TDZ directly.
Acknowledgements This work forms part of the activities of the Centre for Offshore Foundation Systems (COFS), currently supported as a node of the Australian Research Council Centre of Excellence for Geotechnical Science and Engineering and as a Centre of Excellence by the Lloyd's Register Foundation (LRF). Lloyd's Register Foundation helps to protect life and property by supporting engineeringrelated education, public engagement and the application of research. The first author acknowledges her research studentship support from The University of Western Australia (UWA). References Barltrop, N.D.P., Adams, A.J.,1991. Dynamics of Fixed Marine Structures. Butterworth-Heinemann, Marine Technology Directorate. Bea, R.G., Xu, T., Stear, J., Ramos, R., 1999. Wave forces on decks of offshore platforms. J. Waterw. Port Coast. Ocean Eng. 125 (3), 136–144. Bridge, C., 2005. (Ph.D. thesis). Effects of Seabed Interaction on Steel Catenary Risers. University of Surrey, Guildford, UK.
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Bridge, C., Laver, K., Clukey, E., Evans, T., 2004. Steel catenary riser touchdown point vertical interaction models. In: Proceedings of the Offshore Technology Conf., Houston, USA, OTC 16628. DNV-OS-F201, October 2010. Offshore Standard, Dynamic Risers. Det Norske Veritas, Hovik, Norway. DNV-RP-C203, October 2011. Recommended Practice, Fatigue design of offshore steel structures. Det Norske Veritas, Hovik, Norway. Kimiaei, M., Randolph, M.F., Ting, I.,2010. A parametric study on effects of environmental loadings on fatigue life of steel catenary risers (using a nonlinear cyclic riser–soil interaction model). In: Proceedings of the 29th Int. Conf. on Ocean, Offshore and Artic Engineering. Shanghai, China, OMAE2010-21153. Orcina (2011). OrcaFlex User Manual 9.5. 〈www.orcina.com〉,UK. Quéau, L.M., 2014. (Forthcoming Ph.D. thesis). Estimating the Fatigue Damage of Steel Catenary Risers in the Touchdown Zone. The University of Western Australia, Perth, Australia. Quéau, L.M., Kimiaei, M., Randolph, M.F.,2011. Dynamic amplification factors for response analysis of steel catenary risers at touch down areas. In: Proceedings of the 21st Int. Offshore and Polar Engineering Conference. Hawaii, USA, vol. II, pp. 1–8. Quéau, L.M., Kimiaei, M., Randolph, M.F., 2013. Dimensionless groups governing response of steel catenary risers. J. Ocean Eng. 74, 247–259. Quéau, L.M., Kimiaei, M., Randolph, M.F., 2014a. Analytical estimation of static stress range in steel catenary risers at touchdown area and its application with dynamic amplification factors. J. Ocean Eng. 88, 63–80. Quéau, L.M., Kimiaei, M., Randolph, M.F., 2014b. Artificial neural network development for stress analysis of steel catenary risers: sensitivity study and approximation of static stress range. Appl. Ocean Res. 48, 148–161. Quéau, L.M., Kimiaei, M., Randolph, M.F.,2015. Approximation of the maximum dynamic stress range in steel catenary risers using artificial neural networks. Under review. Ruiz-Teran, A.M., Aparicio, A.C., 2006. Dynamic amplification factors in cable-stayed structures. J. Sound and Vib. 300 (2007), 197–216. Shiri, H., Hashemi, H.,2012. Simplified approximation of peak fatigue damage in the touchdown area of steel catenary risers based on seabed soil rigidity. In: Proceedings of the 22nd International Offshore and Polar Engineering Conference (ISOPE). Rhodes, Greece, pp. 287–294. Xia, J., Das, P.K., Karunakaran, D.,2008. A parametric study for the design of steel catenary rser system in deepwater harsh environments. In: Proceedings of the 27th International Conference on Ocean, Offshore and Artic Engineering. Estoril, Portugal, Paper OMAE2008-57179. Zhan, J.P., 2010. (Thesis supervised by Prof. B. J. Leira). Review and Verification of Marine Riser Analysis Programs: Global Response Analysis. Norwegian University of Science and Technology (NTNU), Department of Marine Technology, Trondheim, Norway.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Sensitivity studies of SCR fatigue damage in the touchdown zone using an efficient simplified framework for stress range evaluation Lucile M. Quéau n, Mehrdad Kimiaei, Mark F. Randolph Centre for Offshore Foundation Systems, The University of Western Australia, Crawley, WA 6009, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 14 July 2014 Accepted 20 December 2014 Available online 30 January 2015
Steel catenary risers (SCRs) are widely used in deep water. Several sources of nonlinearities make SCR fatigue design challenging. Limited understanding of the influence of the various input parameters on the structural response of SCRs leads to unnecessarily high conservatism in design. Also, time consuming numerical simulations are usually performed to assess SCR fatigue damage which is inefficient, especially for early design stages. A simplified framework for fatigue analysis of SCRs in the touchdown zone (TDZ) has been developed previously, using artificial neural networks. The approach may be used to efficiently estimate maximum static and dynamic stress ranges in the TDZ, from which the fatigue damage can be deduced. Comparison of the maximum static and dynamic stress changes for a given input motion allows quantification of the dynamic amplification factor (DAF). This paper explores the sensitivity of the maximum dynamic stress ranges and DAF to the key dimensionless groups of input parameters and also certain individual input parameters. The study illustrates the usefulness of the proposed framework in understanding SCR behaviour in the TDZ, providing guidance on optimisation of SCR design from a fatigue perspective. The paper also reflects on the potential benefits of using DAFs for SCR fatigue design. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Steel catenary risers Fatigue Sensitivity analyses Artificial neural network Dynamic amplification factor
1. Introduction Oil and gas developments in deep water commonly use steel catenary risers (SCRs) to transfer fluids between seabed and sea surface as they are a cost effective solution. Under the action of environmental loading a complex interaction is created in the touchdown zone (TDZ—the dynamic area of riser-soil interaction (Bridge, 2005)) and fatigue damage is generated. Fatigue design is a major challenge for SCRs as there is a lack of understanding of the influence of the various input parameters on the fatigue damage, leading to unnecessary conservatism. Fatigue damage is usually evaluated by running time consuming numerical simulations. This approach is inefficient, especially for the early stages of design where optimisation studies are required to find values of input parameters leading to the best performance. A simple approach able to provide quantitative guidance on how the input parameters impact the static and dynamic response of SCRs in the TDZ, thus allowing rapid evaluation of the fatigue damage in the TDZ, would be better suited for initial design. The development of such an approach was indeed encouraged in a recent standard for the design of offshore risers (DNV-OS-F201, 2010).
n
Corresponding author. Fax: þ 61 8 6488 1044. E-mail address: [email protected] (L.M. Quéau).
http://dx.doi.org/10.1016/j.oceaneng.2014.12.038 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
A programme of research (Quéau, 2014; Quéau et al., 2011, 2013, 2014a, 2014b, in preparation) has been undertaken by the authors with the ultimate aim of proposing a simplified fatigue analysis and design framework for SCRs in the TDZ. The original intention was to use the dynamic amplification factor (DAF) approach for dynamic response of SCRs, since it is used widely for simplification of structural dynamic analyses for linear structural systems (e.g. Barltrop and Adams, 1991; Bea et al., 1999; Ruiz-Teran and Aparicio, 2006). DAFs quantify the amplification of stress due to dynamic effects when compared with the static response. The simplification relies on the ability to evaluate the static response and the DAF values through simple methods, and hence deduce the dynamic response. However, SCR behaviour is impacted by geometrical and material nonlinearities in the TDZ and therefore the usefulness of the DAF approach for SCR design was explored first. The study is limited to 2D conditions where the current profile in the sea column, the rotational stiffness at the hang-off point (HOP), the coating and the flow rate of the content are not taken into account. Only the case of a linear soil model, horizontal and flat seabed is investigated here. Also, an illustration of the main input parameters is given in Fig. 1 and the following parameters remain unchanged in the study:
Gravity acceleration (g¼ 9.81 m/s2), Young's modulus and Poisson's ratio for the SCR (E ¼2.12E8 kPa, ν ¼ 0.293), water and steel density (ρwater ¼ 1.025 te/m3; ρsteel ¼7.85 te/m3),
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soil friction coefficient (μ ¼0.5), hydrodynamic coefficients (CD ¼1.2; CA ¼ 1), angular position on the SCR circumference (β ¼01), angle of the motion relative to the hang-off angle (Δθm ¼ θm θHO ¼01, as detailed hereafter).
The loading of SCRs is represented by harmonic motions of the floating vessel through the procedure suggested by Kimiaei et al. (2010), where a time varying sinusoidal motion of the vessel, making an angle θm with the vertical and characterised by heave amplitude (H) and period (T) of the motion (i.e. maximum input velocity of 2πH/T) is applied. The vessel is always a semisubmersible
in this study, with its motion applied parallel to the riser tangential line at the hang-off point (hence Δθm set to zero). This is based on the findings from Kimiaei et al. (2010) that, with a semisubmersible, the tangential component of motion alone could adequately represent the fatigue damage resulting from wave loading in the TDZ. Under these assumptions, a pilot study was performed, defining the DAF for SCRs as: DAF¼Max ΔσTDZ_dyn/Max ΔσTDZ_sta, where Max ΔσTDZ_sta and Max ΔσTDZ_dyn are the maximum stress ranges occurring in the TDZ under static and dynamic loading respectively (Quéau et al., 2011). Since the study indicated that the DAF approach was well suited to SCR fatigue design in the TDZ, the more substantial steps necessary to develop the approach were pursued. The major challenge was to predict DAF values and their sensitivity to key input parameters. As a pre-requisite to the sensitivity studies, dimensional analysis was carried out for SCR systems under harmonic motions to identify the dimensionless groups of input parameters influencing the stress range, and consequently the DAF (Quéau et al., 2013). A summary of the various dimensionless groups is given in Fig. 2, using the dimensionless groups notation introduced in Quéau et al. (2013). The validity of the dimensionless groups was tested through numerical analyses and demonstrating that, under the simplifying assumptions adopted in the study, the maximum stress ranges occurring respectively under dynamic motions (accounting for hydrodynamic and inertia effects as longitudinal and transverse waves travel along the riser) and static motions (no inertia or damping effects) could be expressed as follows: Max Δσ TDZ_dyn H ¼f ; Δθm ; E Δz
Fig. 1. SCR geometry and key parameters: (a) side view of SCR configuration; and (b) Riser cross-section (Quéau et al., 2013).
H T
rffiffiffiffiffiffiffiffiffiffi
ρsteel Do Do p To ks ρ g Δzρsteel ; ;β ; ; ; ν; ; μ; ; C D ; C A ; steel ; E Δz w t E Δz E Δz 2 E ρwater E
ð1Þ
Fig. 2. Flowchart of the ANN framework.
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Table 1 Selected extreme values of the individual input parameters. Varied input parameter
Minimum value
Maximum value
Water depth, Δz Outside diameter, Do Wall thickness, wt
400 m 0.1524 m Do/15 if 0.1524 mr Do o 0.36 m Do/20 if 0.36 m rDo o0.56 m Do/25 if 0.56 m rDo r0.762 m 0 kg/m3
2000 m 0.762 m Do/9 if 0.1524 m r Do r 0.36 m Do/12 if 0.36 mo Do r 0.56 m Do/15 if 0.56 m oDo r0.762 m 1025 kg/m3 if 0.1524 mr Do r 0.46 m 800 kg/m3 if 0.46 mo Do r 0.762 m Such that: θHO ¼201 if 400 mr Δz r950 m θHO ¼171 if 950 m oΔz r 1500 m θHO ¼111 if 1500 m o Δzr 2000 m
Content density, ρcont Horizontal tension component, To
Such that: θHO ¼ 171 if 400 mr Δzo 950 m θHO ¼ 91 if 950 mr Δz o 1500 m θHO ¼ 71 if 1500 m rΔz r 2000 m
Heave amplitude, H Soil stiffness, ks Period of the input motion, T
0.1 m 11.4 kPa 4 s if 0.1 mr Ho 1 m 7 s if 1 mr Ho 3.5 m 10 s if 3.5 m rH o5.5 m 13 s if 5.5 m rH r7.5 m
7.5 m 228 kPa 20 s
Table 2 Selected ranges of the dimensionless groups for SCR stress analysis. Varied dimensionless group
Minimum value ( )
Maximum value ( )
Riser displacement amplitude, π2 ¼ H/Δz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Riser displacement velocity, π4 ¼ðH=TÞ ρsteel =E Riser outside diameter, π5 ¼ Do/Δz Riser outside diameter to wall thickness ratio, π6 ¼ Do/wt Riser unit submerged weight, π7 ¼ p/(EΔz) Riser tension, π9 ¼ To/(EΔz2) Soil stiffness, π11 ¼ ks/E Water depth, π15 ¼ gΔzρsteel/E
5.00E 05 9.62E 07
1.88E 02 1.11E 4
7.62E 05 9 3.92E 13 5.44E 14 5.38E 08 1.45E 04
1.91E 03 25 8.07E 11 4.19E 11 1.08E 06 7.26E 04
Table 3 Parts removed of the design space for dynamic loading. Range of Δz (restricting the range of π15)
Area excluded from the design space for dynamic loading (for any appropriate values of π4, π5, π6, π7, π9, π11)
400 m r Δzr 950 m 950 mo Δz r1500 m 1500 m oΔz r 2000 m
π2 r 6.88E 4 π2 r 1.89E 4 π2 r 1.27E 4
Max Δσ TDZ_sta H ¼f ; Δθ m ; E Δz Do Do p To ks ρ gΔzρsteel ; ; ; ν; ;β ; μ; ; steel ; Δz wt EΔz EΔz2 E ρwater E
ð2Þ
β is the angular position on the SCR circumference, Δz is the Δθm is the angle of the motion relative to the hang-off angle (θHO), μ is the soil friction coefficient, ν is Poisson's ratio, ρsteel and ρwater are the steel and
where
vertical difference between hang-off point and seabed,
water densities, CD and CA are the drag and added mass coefficients, Do are wt, are the riser outer diameter and wall thickness, E is Young's modulus, g is the gravity acceleration, H and T are the heave amplitude and period of the input motion, ks is the soil stiffness, p is the unit submerged weight, and To is the horizontal tension component. The dimensionless groups of parameters impacting the value of DAF are the same as those for the maximum dynamic stress range, because of the DAF definition. Throughout the study, numerical analysis of the SCR system has been carried out using the dynamic
analysis software OrcaFlex (Orcina, 2011). Approximate, but accurate, analytical solutions can evaluate the static response (and maximum static stress range) of an SCR (Quéau et al., 2014a). However, since it is complex to develop analytical solutions able to accurately evaluate the dynamic response of SCRs in time domain, the approach has been to develop artificial neural networks (ANNs), trained using the numerical results from OrcaFlex. This approach was validated initially for the static response (Quéau et al., 2014b) before embarking on the dynamic response. Extensive sensitivity studies were performed to capture SCR behaviour in the TDZ for many thousands of different configurations, loading conditions, riser properties etc. (Quéau et al., 2014a, 2014b, Under review). The sensitivity studies were originally intended to increase the understanding of SCR behaviour and predict the DAF value using an advanced mathematical tool, the artificial neural network (ANN). Since the same amount of computational effort was required for approximating the relationships between either Max ΔσTDZ_dyn or DAF, it was decided to use the ANN tool to capture the relationships between the input dimensionless groups and Max ΔσTDZ_dyn directly. Indeed, this is essentially more convenient for the final user in future applications (the maximum dynamic stress range being directly linked to the fatigue damage). The development of such a framework was performed in Quéau et al. (2014a, 2014b, Under review). The main aim of this paper is to link together all of the previous steps of the study, focusing on two distinct aspects: (i) After presenting the ANN framework (Section 2), it is used to efficiently explore the sensitivity of the maximum stress range in the TDZ to variations of some of the input dimensionless
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Table 4 Performance of the approximations from the ANN framework.
Database size Proportion of cases with errors within:
75% 715%
Reference
groups and individual parameters (Section 3); the results are then used to optimise an example SCR system in respect of the fatigue life (Section 4). (ii) Reflecting back on the initial strategy of combining the static response with a dynamic amplification factor, the ANN framework is used to examine the sensitivity of the DAF to some of the input dimensionless groups and individual parameters (Section 3). The key question to resolve is whether the DAF (together with the static response) is more predictable than the maximum dynamic stress range in the TDZ directly, and thus would offer a more robust design approach, just as commonly used for other types of structural design involving dynamic response.
2. The simplified framework 2.1. Selected range of applicability Under the assumptions of the study, there are respectively seven and eight key input dimensionless groups influencing respectively the maximum stress ranges in the TDZ for SCRs under static and dynamic conditions. These dimensionless groups are as follows (using dimensionless groups notation detailed in Fig. 2 (Quéau et al., 2013)):
Riser displacement, π2 ¼ H/Δz. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Riser displacement velocity, π4 ¼ðH=TÞ ρsteel =E (only for
dynamic motion). Riser outside diameter, π5 ¼Do/Δz. Riser outside diameter to wall thickness ratio, Riser unit submerged weight, π7 ¼p/(EΔz). Riser tension, π9 ¼ To/(EΔz2). Soil stiffness, π11 ¼ks/E. Water depth, π15 ¼ g Δzρsteel/E.
π6 ¼ Do/wt.
To develop a simplified, but broadly applicable framework it was necessary to select wide ranges for these key input dimensionless groups, which define the design space, corresponding to wide ranges of the inherent individual input parameters. The selected ranges are presented in Tables 1 and 2. A series of criteria, referred to as “design criteria” as presented in Table 1, were implemented between some of the input parameters and dimensionless groups to select realistic combinations of dimensionless groups when defining SCR configurations and loading conditions. For instance, small hang-off angles were considered for deep water and larger for shallower water to match with industry practices. The riser displacement characteristics were derived from in-service conditions of a typical semisubmersible vessel under calm to harsh seastates in the Gulf of Mexico (GoM). For the dynamic motion, small parts of the design space had to be excluded since it was not possible at this stage to define an approximation of the relationships between the input dimensionless groups and the maximum normalised stress range in the TDZ with a sufficient level of accuracy in these areas, as discussed later. These areas are summarised in Table 3. Further details on the choice of ranges can be found in Quéau et al. (2014a, 2014b, Under review).
9-ANNs static approximation
9-ANNs dynamic approximation
4 50,000 4 99% 100% Quéau et al., 2014b
4 40,000 4 86% 4 97% Quéau et al., Under review
2.2. Framework characteristics A global ANN approximation for the entire design space was developed for the static loading cases initially, in order to ensure the usefulness of the proposed methods, and the more complex dynamic loading cases were examined afterwards. Ultimately, two approximations were established, namely the “9-ANNs static approximation” (Quéau et al., 2014b) and the “9-ANNs dynamic approximation” (Quéau et al., Under review). Both approximations comprise a total of nine single hidden layer Levenberg–Marquardt back-propagation neural networks, having two activation functions (a bipolar sigmoidal function for the nodes of the hidden layer and an identity function for the output nodes) and different number of neurons in the hidden layer (ranging from 20 for some of these ANNs to 100 for others). Each of these ANNs were trained and tested using large databases to find the weights and biases matrices leading to a good match between approximated results and results from OrcaFlex simulations (Quéau et al., 2014b, Under review). These approximations were developed as standalone MATLAB applications and have been combined into a single standalone MATLAB application, namely “ANN framework”, for convenience.1 The flowchart for the application is given in Fig. 2. All the matrices of weights and biases inherent to the various ANNs are built-in the standalone application for straightforward use. The application uses an Excel™ interface so that a user can simply copy and paste a dataset (i.e. series of dimensionless groups' combinations defining SCR configurations and loading conditions, as per the “Varied dimensionless groups” in Fig. 2) and automatically obtain the maximum stress range corresponding to the static and dynamic loading of the specified SCRs and loading conditions (as per the “Approximated outputs” in Fig. 2). The standalone application performs matrix multiplications based on the input values and the built-in matrices of weights and biases to calculate the maximum stress range results (Quéau et al., 2014b, Under review), and therefore the time taken to develop the approximations does not impact the time necessary to assess the outputs with the ANN framework application. The fatigue damage can then be evaluated directly from the dynamic results. The ANN framework application also reports DAF values, evaluated using the static and dynamic results obtained with the 9-ANNs static and dynamic approximations. 2.3. Performance of the ANN framework The ANN framework application is very efficient, taking about 1 min to estimate the outputs (Max ΔσTDZ_sta, Max ΔσTDZ_dyn and DAF) for 50,000 cases (different SCR configurations with given static or dynamic loading conditions) and without the need for any specialised software. It is difficult to quantify accurately the time saving impact of the ANN framework since the time necessary to obtain stress range results from marine analysis software (such as OrcaFlex, as used here) will vary from user to user, depending on the experience of the user, the performance of the computer used to run the simulations and on the efficiency of pre- and postprocessing the numerical models. The last aspect may be 1
The application may be obtained through the corresponding author.
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streamlined by the use of automation sub-routines (Quéau et al., 2014a, 2014b, Under review). However, four months of calculation were necessary here to obtain the results for the entire database using OrcaFlex running on a high performance computer. The proposed simplified framework represents therefore a tremendous time saving in fatigue design studies, with only limited loss in accuracy. The two approximations provide a good level of accuracy as summarised in Table 4. Excellent accuracy was consistently reached with the 9-ANNs static approximation. With the 9ANNs dynamic approximation, a reasonably good level of accuracy was obtained, although future work (outside the scope of this paper) was recommended to improve the accuracy further in order to increase the robustness of the predictions (Quéau et al.,
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Under review). There is also a restriction on minor parts of the design space for the 9-ANNs dynamic approximation where the results might not be as accurate as for the rest of the design space and therefore it is not recommended to apply the approximation in these areas. This is the case for regions corresponding to low values of imposed displacement relative to the water depth (π2), as shown in Table 3; these regions represent waves that are unlikely to contribute significantly to the fatigue damage (Quéau et al., Under review). The performances indicated in Table 4 are valid when excluding the low π2 values, as presented in Table 3, from the design space. Application of the 9-ANNs dynamic approximation to a series of SCR configurations was shown to predict the fatigue life within 7 15% of the numerical results (Quéau et al., Under review).
Table 5 Selected values to investigate the sensitivity of SCR behaviour to displacement amplitude and velocity.
Constant
Varied
Input parameter/dimensionless group
Value
Do (m) wt (m) p (kN/m) Δz (m) θHO (1) ks (kPa) ρcont (kg/m3) To (kN) π5 ¼Do/Δz π6 ¼Do/wt π7 ¼p/(EΔz) π9 ¼To/(EΔz2) π11 ¼ks/E π15 ¼ gΔzρsteel/E H (m)
0.4572 0.0286 1.31 1225 13 119.7 0 466 3.73E 04 16 5.05E 12 1.47E 12 5.65E 07 4.45E 04 5 values per subplot: (a) 0.1; 0.325; 0.55; 0.775; 1 (b) 1 þ ; 1.625; 2.25; 2.875; 3.5 (c) 3.5 þ ; 4; 4.5; 5; 5.5 (d) 5.5 þ ; 6; 6.5; 7; 7.5 20 values per subplot, equally spaced within the following ranges: (a) 4–20 (b) 7–20 (c) 10–20 (d) 13–20 5 values per subplot as per appropriate H value:
T (s) π2 ¼H/Δz
(a) (b) (c) (d) π4 ¼ðH=TÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
0.08E 3; 0.27E 3; 0.44E 3; 0.63E 3; 0.82E 3 0.82E 3 þ ; 1.33E 3; 1.84E 3; 2.35E 3; 2.86E 3 2.86E 3 þ ; 3.27E 3; 3.67E 3; 4.08E 3; 4.49E 3 4.49E 3 þ ; 4.9E 3; 5.31E 3; 5.71E 3; 6.12E 3
20 values per selected H value as per appropriate H and T values, in the following ranges: (a)
– – – – –
9.62E 7 to 4.81E 6 for H¼ 0.1 m 3.13E 6 to 1.56E 5 for H¼ 0.325 m 5.29E 6 to 2.65E 5 for H¼ 0.55 m 7.46E 6 to 3.73E 5 for H¼0.775 m 9.62E 6 to 4.81E 5 for H¼ 1 m
(b)
– – – – –
9.62E 6 þ to 2.75E 5 for H¼ 1 þ m 1.56E 5 to 4.47E 5 for H¼ 1.625 m 2.16E 5 to 6.19E 5 for H¼ 2.25 m 2.77E 5 to 7.90E 5 for H¼ 2.875 m 3.37E 5 to 9.62E 5 for H¼ 3.5 m
(c)
– – – – –
3.37E 5 þ to 6.73E 5 for H¼3.5 þ m 3.85E 5 to 7.70E 5 for H¼ 4 m 4.33E 5 to 8.66E 5 for H¼ 4.5 m 4.81E 5 to 9.62E 5 for H¼ 5 m 5.29E 5 to 1.06E 4 for H¼ 5.5 m
(d)
– – – – –
5.29E 5 þ to 8.14E 5 for H¼ 5.5 þ m 5.77E 5 to 8.88E 5 for H¼6 m 6.25E 5 to 9.62E 5 for H¼ 6.5 m 6.73E 5 to 1.04E 4 for H¼ 7 m 7.22E 5 to 1.11E 4 for H¼7.5 m
The superscriptsþ and – in 1 þ or 1 for instance, indicate a value close to 1 but higher or lower than 1 by a negligible amount. They are used to select values to reflect of the design criteria between H and T.
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In the rest of the paper, this simplified framework is used for convenience on a smaller part of the design space where the 9-ANNs dynamic approximation underwent more substantial validations (Quéau et al., Under review). This corresponds to water depths of 950 mo Δzr1500 m (and therefore 3.45E4o π15 r5.45E 4), riser displacement π2 41.89E 4 (due to the comment on lower accuracy and fatigue relevance for low riser displacement) and corresponding ranges for π4, π5, π6, π7, π9, π11. 3. Sensitivity studies using the ANN framework The simplified framework is now used to shed light on the sensitivity of the maximum stress range in SCRs in the TDZ and of the DAF to the variation of some of the key input dimensionless groups of parameters. The resulting sensitivities are benchmarked against results from the published literature. There are many ways to vary the key input dimensionless groups and individual parameters and this paper illustrates the usefulness of the ANN framework by considering a selection of examples of individual and simultaneous variations of the key input dimensionless groups and individual input parameters. The first two examples, testing the effect of the displacement characteristics and the soil stiffness on the maximum stress range in the TDZ, aim to demonstrate that the ANN framework can reproduce expected and known trends to validate further its accuracy. Building on from these two examples, two further examples derived from published studies performed using the traditional approach (i.e. numerical simulations) are presented to illustrate the usefulness of the ANN framework in validating published trends for new ranges of the inputs. Throughout this section and wherever possible based on (i) the tested variations, (ii) the selected values for the input parameters and dimensionless groups, and (iii) the design criteria among them, the unvaried dimensionless groups and individual parameters are fixed to their appropriate mean value, except for π7 the value of which is determined based on the water depth and the riser unit submerged weight for an empty riser (for convenience). The term ‘appropriate’ implies that the design criteria between individual input parameters and dimensionless groups, as shown in Table 2, are respected and refers to the mean values accounting for the selected range of water depth: 950 m o Δzr1500 m (or 3.45E 4 o π15 r5.45E 4). Moreover, the reported values of the horizontal component of the tension (To) is assessed by using one of the common catenary equations: To ¼ pΔzsin(θHO)/(1 sin (θHO)). 3.1. Effect of the imposed riser displacement amplitude (π2) and velocity (π4) The effect of the imposed riser displacement amplitude (π2) and velocity (π4) on SCR behaviour in the TDZ is investigated first. For this purpose, H and T were varied within the four selected ranges of H and corresponding T values. Table 5 summarises the selected values of the individual input parameters and dimensionless groups that were used for this example. A series of design charts showing the sensitivity of Max ΔσTDZ_dyn/E and DAF to variations of π2 and π4 were established, as illustrated in Figs. 3 and 4 respectively. The range of π4 was normalised in the subplots for convenience, using the extreme maximum (max) and minimum (min) values shown in Table 5 (so that Normalised π4 ¼(2* π4 - max - min)/(max - min)). Results are discussed next, for the maximum stress range in the TDZ first and then for the DAF. 3.1.1. Maximum stress range in the TDZ Increasing the velocity (π4) by keeping the displacement amplitude constant (π2), thereby reducing the period of the imposed
displacement (T), leads to an increase of Max ΔσTDZ_dyn/E. Also, the higher the displacement amplitude, the higher the maximum stress range in the TDZ (i.e. increase of fatigue damage) since it generates higher variations of the curvature in the TDZ. These trends are observed for the four ranges of H and corresponding T values selected and show that no resonance effects were detected for the SCR configuration under study.
3.1.2. DAF The values of maximum stress range found under static loading (Max ΔσTDZ_sta/E) using the simplified framework are shown in Table 6. They were used to calculate the DAF values from Max ΔσTDZ_dyn/E results presented in Fig. 3. Fig. 4 indicates that the DAF increases with increasing velocity of the imposed displacement. Also, it tends to confirm an interesting trend of the DAF sensitivity that was already noted at the pilot study stage of this research (Quéau et al., 2011). For low displacement amplitudes, DAF varies inversely with the displacement amplitude, whereas it increases with increasing displacement amplitude for higher heave amplitude values; Fig. 4b captures the transition between these opposing trends. Interestingly, DAF values do not decrease down to unity in any of the plots presented in Fig. 4, indicating that the SCR under study did not reach its static response for the selected displacements and wave periods. To some extent, this is consistent with the results obtained during the pilot study (Quéau et al., 2011). Even though the selected values for the input parameters and dimensionless groups are not all identical in this paper to those selected for the pilot study, DAF values of unity were only reached in the pilot study when considering imposed displacements with a period (T) greater than 20 s. That limit is beyond the highest limit selected for T here and also when defining the ANN framework (see Table 1 and Quéau et al., 2014a, 2014b, Under review). The similarity of the observed trends when using the ANN framework to those of the pilot study, when DAF sensitivity was investigated using results from numerical simulations, further validates the framework. 3.2. Effect of the soil stiffness (π11) for various imposed displacement characteristics This example focuses on the effect of the soil stiffness (π11) on SCR behaviour in the TDZ for various characteristics of the imposed displacement. The values of the individual input parameters and dimensionless groups used for this purpose are summarised in Table 7 and the design charts are shown in Figs. 5 and 6 for Max ΔσTDZ_dyn/E and DAF respectively. 3.2.1. Maximum stress range in the TDZ The overall trend of the results presented in Fig. 5 is for increasing values of Max ΔσTDZ_dyn/E with increasing soil stiffness, which is consistent with other published work (e.g. Bridge et al., 2004; Quéau et al., 2011). The trend results from higher contact forces and greater curvature variation in the TDZ with increasing soil stiffness. There are minor fluctuations in Fig. 5d where higher soil stiffness locally gives lower values of Max ΔσTDZ_dyn/E for low values of displacement velocity. This may however indicate local inaccuracy of the ANN framework rather than an indication of a different trend for the results in that area. As is true for any simplified approach (DNV-OS-F201, 2010), engineering judgement is required when interpreting results from the proposed framework to differentiate between results indicating a change in SCR behaviour and those affected by slight inaccuracies of the simplified framework.
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3.2.2. DAF DAF increases with increasing velocity of the imposed displacement, as observed for all selected values of soil stiffness and
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displacement amplitude (see Fig. 6). Except for high velocity values (i.e. low period of imposed displacement), DAF is almost insensitive to the soil stiffness. This confirms the trend observed in
Fig. 3. Sensitivity of Max ΔσTDZ_dyn/E to the variation of the loading amplitude (π2) and velocity (π4), accounting for the design criteria between H and T: (a) 0.1 m rH o1 m; (b) 1 mr Ho 3.5 m; (c) 3.5 m r Ho5.5 m; and (d) 5.5 mr Hr 7.5 m.
Fig. 4. Sensitivity of DAF to the variation of the loading amplitude (π2) and velocity (π4), accounting for the design criteria between H and T: (a) 0.1 m rHo 1 m; (b) 1 mr Ho 3.5 m; (c) 3.5 m r Ho5.5 m; and (d) 5.5 mr Hr 7.5 m.
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Quéau et al. (2011) with other values of input parameters and dimensionless groups (e.g. different values of water depth, riser diameter and wall thickness, hang-off angle etc…).
Table 6 Static stress range results necessary to explore DAF sensitivity to displacement amplitude and velocity. H (m)
π2 ¼ H/Δz
Max ΔσTDZ_sta/E
0.100 0.325 0.550 0.775 1.000 1.625 2.250 2.875 3.500 4.000 4.500 5.000 5.500 6.000 6.500 7.000 7.500
8.16E 05 2.65E 04 4.49E 04 6.33E 04 8.16E 04 1.33E 03 1.84E 03 2.35E 03 2.86E 03 3.27E 03 3.67E 03 4.08E 03 4.49E 03 4.90E 03 5.31E 03 5.71E 03 6.12E 03
9.80E 06 3.17E 05 5.33E 05 7.46E 05 9.57E 05 1.51E 04 2.02E 04 2.48E 04 2.90E 04 3.21E 04 3.48E 04 3.74E 04 3.97E 04 4.19E 04 4.38E 04 4.56E 04 4.72E 04
3.3. Effect of the water depth (π15) for various hang-off angles and imposed displacement characteristics In this example the effect of increasing water depth (π15) for a constant value of hang-off angle is investigated. Various values of hang-off angle and imposed displacement characteristics are considered. This example was selected to compare the results found with the simplified framework and for the selected ranges of parameters against the results reported by Zhan (2010), established using numerical simulations. The selected values of input parameters and dimensionless groups are presented in Table 8 and the results are shown in Figs. 7–9. The hang-off angle is measured from the vertical.
3.3.1. Maximum stress range in the TDZ The static results are presented in Fig. 7 and indicate that increasing water depth leads to a reduction of Max ΔσTDZ_sta/E. Also, higher hang-off angle values result in lower values of Max ΔσTDZ_sta/E due to increased tension in the TDZ (π9), thereby reducing the curvature variation in the TDZ. The marked change in gradient observed between the two lowest values of water depth (particularly for low hang-off angles) may indicate slight inaccuracy of the ANN approximation rather than reflecting true SCR behaviour. Indeed, estimation of the maximum stress range in
Table 7 Selected values to investigate the sensitivity of SCR behaviour to the variation of the soil stiffness for various displacement characteristics.
Constant
Varied
Input parameter/dimensionless group
Value
ρcont (kg/m3) Do (m) wt (m) p (kN/m) Δz (m) θHO (1) To (kN) π5 ¼ Do/Δz π6 ¼ Do/wt π7 ¼ p/(EΔz) π9 ¼ To/(EΔz2) π15 ¼ gΔzρsteel/E ks (kPa) H (m)
0 0.4572 0.0286 1.31 1225 13 466 3.73E 04 16 5.05E 12 1.47E 12 4.45E 04 5 values in each subplot: 11,4; 65.55; 119.7; 173.85; 228 1 value per subplot: (a) 0.55 (b) 2.25 (c) 4.5 (d) 6.5
T (s)
20 values per subplot, equally spaced within the following ranges: (a) 4–20 (b) 7–20 (c) 10–20 (d) 13–20
π2 ¼ H/Δz
1 value per subplot as per appropriate H value: (a) 4.49E 4 (b) 1.84E 3 (c) 3.67E 3 (d) 5.31E 3
π4 ¼ ðH=TÞ
π11 ¼ ks/E
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
20 values per subplot, equally spaced within the following ranges: (a) 5.29E 6 to 2.65E 5 (b) 2.16E 5 to 6.19E 5 (c) 4.33E 5 to 8.66E 5 (d) 6.25E 5 to 9.62E 5 5 values in each subplot: 5.38E 8; 3.09E 7; 5.65E 7; 8.20E 7; 1.08E 6
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Fig. 5. Sensitivity of Max ΔσTDZ_dyn/E to the variation of the displacement velocity (π4) for various soil stiffnesses (π11) and displacement amplitudes: (a) π2 ¼0.44E 3; (b) π2 ¼ 1.84E 3; (c) π2 ¼ 3.67E 3; and (d) π2 ¼5.31E 3.
Fig. 6. Sensitivity of DAF to the variation of the displacement velocity (π4) for various soil stiffnesses (π11) and displacement amplitudes: (a) π2 ¼0.44E 3; (b) π2 ¼1.84E 3; (c) π2 ¼3.67E 3; and (d) π2 ¼ 5.31E 3.
the TDZ for the extreme values of inputs may not be as accurately captured as for intermediate input values, which is inherent to the response surface method.
For dynamic results, Fig. 8 shows that increasing water depth generally results in a slight reduction of the dynamic stress range (Fig. 8a–c) although an opposite trend is found in Fig. 8d for the
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Table 8 Selected values to investigate the sensitivity of SCR behaviour to the variation of water depth for various hang-off angles and displacement characteristics.
Constant
Varied
Input parameter/dimensionless group
Value
ks (kPa) ρcont (kg/m3) Do (m) wt (m) p (kN/m) π6 ¼Do/wt π11 ¼ks/E Δz (m) θHO (1) H (m)
119.7 0 0.4572 0.0286 1.31 16 5.65E 07 20 values equally spaced within the following range: 950–1500 5 values in each subplot: 9; 11; 13; 15; 17 1 value per subplot: (a) (b) (c) (d)
T (s)
1 value per subplot: (a) (b) (c) (d)
12 13.5 15 16.5
varying as per selected θHO and Δz values, within the following overall range: 231–813 20 values per selected H value, equally spaced within the following ranges: (a) 3.67E 4 to 5.79E 4 (b) 1.50E 3 to 2.37E 3 (c) 3.00E 3 to 4.74E 3 (d) 4.33E 3 to 6.84E 3
To (kN) π2 ¼H/Δz
π4 ¼ðH=TÞ
0.55 2.25 4.5 6.5
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
π5 ¼Do/Δz π7 ¼p/(EΔz) π9 ¼To/(EΔz2) π15 ¼ gΔzρsteel/E
1 value per subplot: (a) 8.82E 6 (b) 3.21E 5 (c) 5.77E 5 (d) 7.58E 5 20 values depending on the value of Δz and equally spaced within the 20 values depending on the value of Δz and equally spaced within the varying as per selected θHO and Δz values, within the following overall 20 values depending on the value of Δz and equally spaced within the
highest displacement amplitude, H¼ 6.5 m. Also, lower hang-off angles lead to higher values of Max ΔσTDZ_dyn/E for all the selected cases expect for H ¼6.5 m where the opposite trend is observed. The trend illustrated in Fig. 8d is consistent with the results from Xia et al. (2008) who found that, for a given water depth, increasing the hang-off angle increased the fatigue damage. Further work is needed to ensure that the abrupt variations exhibited by the results in Fig. 8 reflect true physical behaviour of SCRs in the TDZ and are not the result of the ANN framework. For both static and dynamic conditions, the variation of maximum stress range in the TDZ with water depth is quite limited, showing a rather weak sensitivity for a given hang-off angle. These results are consistent with the results from Zhan (2010), who worked with bending moment (which dominates the contribution to SCR stresses in the TDZ (Shiri and Hashemi, 2012)) and found that the dynamic bending moment envelope did not vary much with changing water depth.
3.3.2. DAF The DAF results are shown in Fig. 9. In general, the DAF increases with increasing water depth and smaller hang-off angles generate less dynamic amplification (Fig. 9a, c and d); a slight opposite trend is found in Fig. 9b, although with low sensitivity of the DAF to the hang-off angle.
following range: 3.05E 4 to 4.81E 4 following range: 8.75E 4 to 1.38E 3 range: 7.65E 13 to 2.69E 12 following range: 3.45E 4 to 5.45E 4
3.4. Effect of the outside diameter (Do) for various wall thicknesses and imposed displacement characteristics This example was inspired by a parametric study performed by Xia et al. (2008) where a SCR base case was established to test the effect of the internal diameter on SCR behaviour in the TDZ. For this purpose, the same loading was applied to the SCR base case for fixed values of hang-off angle (θHO) and wall thickness (wt) but various values of the internal diameter. For the constant wall thickness this resulted in varying the outside diameter (Do), the submerged weight (p) and the horizontal tension component (To) to keep the hang-off angle unchanged. Here, the sensitivity of SCR behaviour to variations of the outside diameter is investigated by considering several values of the wall thickness and also for various displacement characteristics (and corresponding values of p and To), therefore exploring wider ranges of the input parameters. The values of dimensionless groups and individual input parameters used for this purpose are shown in Table 9. Results are presented in Figs. 10–12 and are discussed below.
3.4.1. Maximum stress range in the TDZ In order to compare the results against the findings from Xia et al. (2008), the effect of the variation of the outside diameter on the maximum stress range occurring in the TDZ is illustrated for static and dynamic loading in Figs. 10 and 11 respectively.
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Fig. 7. Sensitivity of Max ΔσTDZ_sta/E to the variation of the water depth (π15) for various hang-off angles (θHO) and displacement characteristics: (a) H¼ 0.55 m; (b) H¼2.25 m; (c) H ¼4.50 m; and (d) H¼ 6.50 m.
Fig. 8. Sensitivity of Max ΔσTDZ_dyn/E to the variation of the water depth (π15) for various hang-off angles (θHO) and displacement characteristics: (a) H¼ 0.55 m and T ¼12 s; (b) H¼2.25 m and T ¼ 13.5 s; (c) H¼4.50 m and T ¼ 15 s; and (d) H¼6.50 m and T ¼ 16.5 s.
In general, increasing the outside diameter for a given wall thickness tends to increase Max ΔσTDZ_sta/E, which is consistent with the results from Xia et al. (2008). However, it seems that for low values of the displacement amplitude a slight opposite trend
may be found (Fig. 10a). Greater wall thickness systematically leads to higher Max ΔσTDZ_sta/E in this example since it results in higher value of submerged weight (π7) and therefore an increase of curvature in the TDZ.
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Fig. 9. Sensitivity of DAF to the variation of the water depth (π15) for various hang-off angles (θHO) and displacement characteristics: (a) H¼0.55 m and T ¼12 s; (b) H¼ 2.25 m and T ¼ 13.5 s; (c) H¼ 4.50 m and T ¼ 15 s; and (d) H¼ 6.50 m and T¼ 16.5 s.
Regarding dynamic loading, Max ΔσTDZ_dyn/E also increases with increasing outside diameter for most of the selected cases, although it decreases for high displacement amplitude and low wall thickness (Fig. 11d). In their case study, Xia et al. (2008) found that the fatigue damage for dynamic loading decreased with increasing outside diameter for a given wall thickness. Further work is therefore needed to validate the observed trends and find the values of input parameters that lead to a change of trends. Another complex behaviour is also noted depending on the value of the displacement amplitude: for low displacement amplitude, the higher the wall thickness, the greater the fatigue damage (Fig. 11a and b), whereas the reverse is true for higher values of displacement amplitude (Fig. 11c and d). This illustrates the usefulness of the simplified framework for the early stages of design where designers can optimise values of wall thickness and outside diameter for the waves that contribute most to the overall fatigue damage under the given wave scatter diagram.
ranges of input parameters and dimensionless groups. The examples have illustrated that conclusions obtained from a base case or for small ranges of input parameters and dimensionless groups cannot always be generalised, complicating general quantification of the influence of particular dimensionless groups on the maximum stress range in the TDZ. This reinforces the need for a simplified method that can conveniently assist screening tasks in the early design stages, providing designers with a tool to optimise values of the dimensionless groups within their control (e.g. riser properties) for given values of dimensionless groups that are outside their control (e.g. environmental conditions). In addition, for the selected examples it was shown that DAF values could vary up to values exceeding 16. Overall, the DAF patterns found in the design charts were no simpler than the patterns for the maximum stress range values. This supports the choice to develop an ANN framework to predict Max ΔσTDZ_dyn/E directly rather than to predict DAF values (Quéau et al., Under review).
3.4.2. DAF Based on the maximum stress ranges in the TDZ, DAF sensitivity to the variation of outside diameter depends on the imposed displacement characteristics. For low displacement amplitude, DAF increases with increasing outside diameter and is higher for greater wall thickness values (Fig. 12a and b). However, for higher displacement amplitudes, DAF decreases with increasing outside diameter and lower wall thicknesses lead to higher DAF values (Fig. 12c and d). Large DAF values in excess of 16 are observed for the highest selected displacement amplitude, the lowest selected wall thickness and the lowest selected outside diameter. The four examples selected in this section were used to compare the trends observed when using the proposed simplified framework against published results obtained with different methods, validating further the usefulness of the framework. It also enabled the sensitivity of SCR behaviour in the TDZ to be explored for wider
4. Optimisation of the fatigue life using the ANN framework This section illustrates application of the ANN framework for SCR fatigue design by performing a simple screening study typical of initial design. The case study is based on deterministic fatigue design of an inservice SCR in the GoM, namely ‘Base Case 1’ (BC1), characteristics of which are presented in Table 10. A simplified wave scatter table was used and 15 load cases (LCs) characterised by a sinusoidal vessel motion (displacement of amplitude H and period T) were selected to represent the loading conditions. These LCs were derived from a selection of 15 waves from a sample wave scatter table for GoM, with wave heights, periods and number of occurrences presented in Table 11, which represent over 95% of the waves occurring over a 20 year period. The values of the corresponding heave amplitudes were assessed based on RAO
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Table 9 Selected values to investigate the sensitivity of SCR behaviour to the variation of outside diameter for various wall thicknesses and displacement characteristics. Input parameter/dimensionless group Constant Δz (m) θHO (1) ks (kPa) ρcont (kg/m3) π11 ¼ ks/E π15 ¼ gΔzρsteel/E Varied Do (m)
Value
1225 13 119.7 0 5.65E 07 4.45E 04 20 values per selected wt value, equally spaced within the following ranges (selected to respect design criteria between value of Do and π6): – – – – –
wt (m) H (m)
0.152 to 0.24 for wt ¼ 0.016 m 0.162 to 0.27 for wt ¼ 0.018 m 0.18 to 0.3 for wt ¼ 0.02 m 0.198 to 0.33 for wt ¼ 0.022 m 0.216 to 0.36 for wt ¼ 0.024 m
5 values in each subplot: 0.016; 0.018; 0.02; 0.022; 0.024 1 value per subplot: (a) (b) (c) (d)
T (s)
1 value per subplot: (a) (b) (c) (d)
To (kN)
121 to 146 for wt ¼0.016 m 149 to 185 for wt ¼ 0.018 m 184 to 229 for wt ¼ 0.02 m 223 to 277 for wt ¼ 0.022 m 265 to 330 for wt ¼0.024 m
20 values per selected wt value, equally spaced within the following ranges:
p (kN/m)
– – – – – π2 ¼ H/Δz
0.34 0.42 0.52 0.64 0.75
to to to to to
0.41 for wt ¼ 0.016 m 0.52 for wt ¼0.018 m 0.64 for wt ¼0.02 m 0.78 for wt ¼0.022 m 0.93 for wt ¼0.024 m
1 value per subplot as per appropriate H value: (a) (b) (c) (d) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
π5 ¼ Do/Δz
4.49E 4 1.84E 3 3.67E 3 5.31E 3
1 value per subplot as per appropriate H and T value: (a) (b) (c) (d)
8.82E 6 3.21E 5 5.77E 5 7.58E 5
20 values per selected wt value, equally spaced within the following ranges: – – – – –
π6 ¼ Do/wt
12 13.5 15 16.5
20 values per selected wt value, equally spaced within the following ranges: – – – – –
π4 ¼ ðH=TÞ
0.55 2.25 4.5 6.5
1.23E 4 1.32E 4 1.47E 4 1.62E 4 1.76E 4
to 1.96E 4 for wt ¼ 0.016 m to 2.20E 4 for wt ¼ 0.018 m to 2.45E 4 for wt ¼ 0.02 m to 2.69E 4 for wt ¼ 0.022 m to 2.94E 4 for wt ¼ 0.024 m
20 values per selected wt value, equally spaced within the following ranges:
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Table 9 (continued ) Input parameter/dimensionless group
Value
– – – – – π7 ¼ p/(EΔz)
20 values per selected wt value, equally spaced within the following ranges: – – – – –
π9 ¼ To/(EΔz2)
9.4 to 15 for wt ¼ 0.016 m 9 to 15 for wt ¼ 0.018 m 9 to 15 for wt ¼ 0.02 m 9 to 15 for wt ¼ 0.022 m 9 to 15 for wt ¼ 0.024 m
1.31E 12 to 1.59E 12 for wt ¼ 0.016 m 1.62E 12 to 2.01E 12 for wt ¼ 0.018 m 2.00E 12 to 2.48E 12 for wt ¼0.02 m 2.41E 12 to 3.00E 12 for wt ¼ 0.022 m 2.87E 12 to 3.57E 12 for wt ¼ 0.024 m
20 values per selected wt value, equally spaced within the following ranges: – – – – –
3.82E 13 to 4.60E 13 for wt ¼ 0.016 m 4.69E 13 to 5.83E 13 for wt ¼ 0.018 m 5.79E 13 to 7.20E 13 for wt ¼ 0.02 m 7.01E 13 to 8.71E 13 for wt ¼0.022 m 8.34E 13 to 1.04E 12 for wt ¼0.024 m
Fig. 10. Sensitivity of Max ΔσTDZ_sta/E to the variation of outside diameter (Do) for various wall thicknesses (wt) and displacement characteristics: (a) H¼ 0.55 m; (b) H¼ 2.25 m; (c) H¼ 4.50 m; and (d) H¼ 6.50 m.
(response amplitude operator) tables of the semisubmersible following the procedure proposed by Kimiaei et al. (2010). It is assumed that a designer wishes to optimise the outside diameter (Do), outside diameter over wall thickness ratio (π6) and hang-off angle (θHO) for the riser presented in BC1 in order to improve the fatigue life. The location of the SCR is fixed so that the loading conditions (displacement amplitude and velocity, π2 and π4
respectively), the soil stiffness (π11) and the water depth (π15) are the same as for BC1 and also the content density (ρcont) is fixed. The only parameters in the designers' control are therefore the following dimensionless groups: π5, π6 and π9 (π7 varying accordingly). The example sensitivity studies in the previous section demonstrated that loading conditions might lead to different choices of recommended values of θHO, Do and wt in order to decrease the
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Fig. 11. Sensitivity of Max ΔσTDZ_dyn/E to the variation of outside diameter (Do) for various wall thicknesses (wt) and displacement characteristics: (a) H¼ 0.55 m and T ¼12 s; (b) H¼2.25 m and T ¼ 13.5 s; (c) H¼4.50 m and T ¼ 15 s; and (d) H¼6.50 m and T ¼ 16.5 s.
Fig. 12. Sensitivity of DAF to the variation of outside diameter (Do) for various wall thicknesses (wt) and displacement characteristics: (a) H¼ 0.55 m and T ¼12 s; (b) H¼ 2.25 m and T ¼13.5 s; (c) H¼ 4.50 m and T ¼15 s; and (d) H¼ 6.50 m and T ¼16.5 s.
maximum dynamic stress range in the TDZ. The fatigue life is based on the cumulative damage from each LC of a wave scatter diagram, and the damage is non-linearly related to the maximum dynamic stress
range in the TDZ through the allowable number of occurrence. It is therefore necessary to find the combination of θHO, Do and π6 values leading to the least overall damage to find the optimum fatigue life.
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For the sake of simplicity, it is assumed that the designer has the choice between three values for each of the inputs, as follows:
Table 10 SCR base cases characteristics. Input parameter/dimensionless group
BC1
Do (m) wt (m) p (kN/m) Δz (m) θHO (1) ks (kPa) ρcont (kg/m3) To (kN) π5 ¼ Do/Δz π6 ¼ Do/wt π7 ¼ p/(EΔz) π9 ¼ To/(EΔz2) π11 ¼ ks/E π15 ¼ gΔzρsteel/E
0.228 ( 9 in.) 0.025 ( 1 in.) 0.82 982 9.8 22.8 0 164 2.32E 04 9.12 3.92E 12 8.05E 13 1.08E 07 3.57E 04
Table 11 Loading conditions for the case studies. Period Number of π2 ¼H/Δz Load Wave Heave occurrence Case height Amplitude T (s) for 20 years H (m) (LC) (m) LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10 LC11 LC12 LC13 LC14 LC15 a
1 1 1 1 3 3 3 8 13 8 8 13 18 13 18
0.004 0.15 0.25 0.39 0.44 0.76 1.17 1.17 1.91 2.02 3.12 3.29 4.55 5.06 7.01
3 8 13 18 8 13 18 8 8 13 18 13 13 18 18
156,673,082 10,946,661 1,132,827 305,883 2,795,859 130,073 16,564 68,228 2073 7409 607 647 92 47 4
4.39E 06a 1.49E 04a 2.58E 04 3.97E 04 4.48E 04 7.73E 04 1.19E 03 1.19E 03 1.94E 03 2.06E 03 3.17E 03 3.35E 03 4.64E 03 5.16E 03 7.14E 03
π4 ¼ ðH=TÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρsteel =E
2.76E 07 3.53E 06 3.75E 06 4.16E 06 1.06E 05 1.12E 05 1.25E 05 2.82E 05 4.58E 05 3.00E 05 3.33E 05 4.87E 05 6.74E 05 5.41E 05 7.50E 05
Value of π2 outside the selected range of applicability of the ANN framework.
θHO can take a value of 9.81 (as for BC1), 131 or 171. Do can take a value of 0.228 m (as for BC1), 0.25 m or 0.35 m. π6 can take a value of 9.12 (as for BC1), 12 or 15. Since there are three possible choices for each of three inputs, 27 possible designs result for the SCR. The ANN framework is applied to investigate the fatigue lives of the SCRs formed by the 27 combinations of inputs, with results shown in Table 12. Fatigue results are assessed using the DNV S–N curve type D for seawater (DNV-RP-C203, 2011). Under those circumstances, by keeping Do and π6 to their values in BC1 and increasing θHO to 171 the fatigue life can be increased from 163 to 722 years, improving fatigue performance by a factor greater than 4. This illustrates how the proposed ANN framework can assist screening design tasks in finding combinations of input parameters that optimise fatigue performance, although other practical considerations may affect the final selection of input parameters. In practice, the ANN framework may be used in conjunction with optimisation software in order to consider many more choices for the input parameters.
5. Conclusions This paper has illustrated the usefulness of the simplified framework based on artificial neural networks (ANNs) developed previously (Quéau et al., 2014b, Under review) for SCR design with respect to fatigue in the TDZ. A series of sensitivity studies were performed by varying a selection of dimensionless groups and individual input parameters to illustrate the trends in SCR behaviour. Estimated trends were benchmarked against published results to validate further the accuracy of the ANN framework. The sensitivity studies explored wider ranges of the inputs and showed how misinterpretations can arise from generalising the trends obtained from more restricted studies, since
Table 12 Screening of SCR characteristics for fatigue performance. Combination
Do (m)
π6 ¼ Do/wt
θHO (1)
π5 ¼ Do/Δz
π7 ¼ p/(EΔz)
π9 ¼To/(EΔz2)
p (kN/m)
To (kN)
Fatigue life (yr)
1 (BC1) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35 0.228 0.25 0.35
9.12 9.12 9.12 12 12 12 15 15 15 9.12 9.12 9.12 12 12 12 15 15 15 9.12 9.12 9.12 12 12 12 15 15 15
9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 13 13 13 13 13 13 13 13 13 17 17 17 17 17 17 17 17 17
2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04 2.32E 04 2.55E 04 3.56E 04
3.92E 12 4.72E 12 9.25E 12 2.64E 12 3.18E 12 6.23E 12 1.79E 12 2.15E 12 4.21E 12 3.92E 12 4.72E 12 9.25E 12 2.64E 12 3.18E 12 6.23E 12 1.79E 12 2.15E 12 4.21E 12 3.92E 12 4.72E 12 9.25E 12 2.64E 12 3.18E 12 6.23E 12 1.79E 12 2.15E 12 4.21E 12
8.05E 13 9.68E 13 1.90E 12 5.42E 13 6.52E 13 1.28E 12 3.66E 13 4.41E 13 8.64E 13 1.14E 12 1.37E 12 2.68E 12 7.67E 13 9.22E 13 1.81E 12 5.19E 13 6.23E 13 1.22E 12 1.62E 12 1.95E 12 3.82E 12 1.09E 12 1.31E 12 2.57E 12 7.38E 13 8.87E 13 1.74E 12
0.82 0.98 1.93 0.55 0.66 1.30 0.37 0.45 0.88 0.82 0.98 1.93 0.55 0.66 1.30 0.37 0.45 0.88 0.82 0.98 1.93 0.55 0.66 1.30 0.37 0.45 0.88
164 198 388 111 133 261 75 90 176 233 280 549 157 188 369 106 127 250 331 398 781 223 268 526 151 181 356
163 169 207 134 129 146 113 107 111 313 322 384 216 210 229 157 150 148 722 708 645 400 403 393 226 233 221
L.M. Quéau et al. / Ocean Engineering 96 (2015) 295–311
the effect of certain inputs on the maximum stress range in the TDZ may depend on the value of other inputs. For this reason, and because the fatigue life depends on the cumulative damage for a given wave scatter diagram, optimisation of an SCR design with respect to fatigue must consider the overall level of damage rather than individual load cases. The usefulness of the ANN framework for such optimisation was illustrated by a simple example. The sensitivity of DAF values, quantifying the dynamic amplification compared with static results, was also investigated. It was shown that values of DAF as high as 16 could occur with the selected examples and that the DAF pattern was no easier to predict than the pattern of the maximum dynamic stress range in the TDZ directly.
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