Sensitivity analysis of jacket-type offshore platforms under extreme waves

Journal of Constructional Steel Research 83 (2013) 147–155

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Journal of Constructional Steel Research

Sensitivity analysis of jacket-type offshore platforms under extreme waves Mona Hezarjaribi a, M.R. Bahaari a, Vahid Bagheri b,⁎, Hossein Ebrahimian b a b

University of Tehran, Tehran, Iran Sharif University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 20 December 2011 Accepted 25 January 2013 Available online 1 March 2013 Keywords: Jacket platforms Sensitivity analysis Incremental Wave Analysis (IWA) Wave loading Pile–Soil–Structure Interaction (PSSI)

a b s t r a c t Jacket-type offshore platforms play an important role in oil and gas industries in shallow and intermediate water depths such as Persian Gulf region. Such important structures need accurate considerations in analysis, design and assessment procedures. In this paper, nonlinear response of jacket-type platforms against extreme waves is examined utilizing sensitivity analyses. Results of this paper can reduce the number of random variables and consequently the computational effort in reliability analysis of jacket platforms, noticeably. Effects of foundation modeling have been neglected in majority of researches on the response of jacket platforms against wave loads. As nonlinear response of the pile foundation is one of the most important sources of potential nonlinearity in the response of offshore platforms, in this study, a powerful model which is able to consider Pile–Soil–Structure Interaction (PSSI) is employed. Therefore, PSSI parameters as well as other parameters such as uncertainties in the prediction of the wave force on jacket structure and uncertainties in structural model are utilized in sensitivity analyses. In this research, pushover methods as well as an advanced approach named “Incremental Wave Analysis (IWA)” are employed. Consequently, collapse prevention limit state of jacket platforms is investigated through different outcomes of pushover and IWA methods including Reserve Strength Ratio, ultimate capacity, collapse displacement and Collapse Wave Height indicators. In order to consider the effects of correlation between random variables, a robust method of sensitivity analysis named correlation coefficient approach is also employed. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction As offshore structures require more critical and complex designs, the need for accurate approaches to evaluate uncertainty and variability in computer models, loads, geometry, and material properties has increased significantly. For problems in which randomness is relatively small, it is evident that a deterministic model is adequate. However, when the level of uncertainty and importance of structure is high, probabilistic approaches should be employed for system analysis and design. One of the fundamental steps in the structural reliability analysis of structures is to determine the significance of random variables, and how they influence the structural response which can be acquired by sensitivity analyses. The sensitivity analysis can answer the basic question “which variables are the most important and should be used in probabilistic analyses?”. The topic of sensitivity analysis for jacket type offshore platforms against wave loading has been widely considered by different researchers. Sunder and Connor [1] investigated the sensitivity of steel jacket offshore platforms to environmental wave loading utilizing two simplified numerical models under rigid foundation conditions. ⁎ Corresponding author. Tel.: +98 1713352976. E-mail address: [email protected] (V. Bagheri). 0143-974X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jcsr.2013.01.013

They studied the effects of wave height, wave period, drag and inertia coefficients, mass and hysteretic structural damping. Hahn [2] used a simplified model in order to examine the effects of inertia and drag force components, current velocity, fluid–structure interaction, random phase angles and wave cancellation. Haver et al. [3] investigated the sensitivity of the annual failure probability to the selected airgap and current design profile. They demonstrated that the airgap parameter is a crucial parameter regarding the annual probability of structural failure. With regard to sensitivity analyses of jacket type platforms, the effects of foundation modeling have been neglected in majority of researches on the response of jacket platforms against wave loads. As nonlinear response of the pile foundation is the most important source of potential nonlinearity in the response of offshore platforms, it is clear that a more powerful model, which is able to consider Pile– Soil–Structure Interaction (PSSI), should be employed. Owjnc [4] studied the sensitivity of the overall dynamic response of the deep water platforms to the variation of the soil characteristics and to the effect of the axial forces of the members utilizing a new formulation. It was illustrated that the overall dynamic responses of a deep water structure can be very sensitive to the foundation model. Kenji Kawano and Katta Venkatammana [5] also conducted dynamic analysis of large offshore structures utilizing the impedance function

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model for the soil–pile foundation system. Moreover, several studies by Makris et al. [6], Mylonakis and Gazetas [7], Guin and Banerjee [8] have focused on PSSI analyses. Bea [9] performed a series of static pushover analyses on a fixed offshore platform and found that the first nine nonlinear events were concentrated in the foundation piles. Moan et al. [10] demonstrated that the choice of pile/soil modeling method can affect the load distribution and failure mode in the structural model. HSE [11] concludes that for ductile jacket platforms, considering nonlinear foundation model results in a significant increase of the lateral displacement of the deck. The effect on the capacity to carry lateral load is, however, small. Although considerable researches on the PSSI have been conducted, most of the earlier studies are based on the assumption that the superstructures are simplified as a block mass or as a series of lumped masses. Therefore, in this paper, a 3-D model of SPD2 jacket platform located in Persian Gulf is utilized to increase the accuracy of the model. In addition, majority of earlier studies have been conducted by means of simple approaches for sensitivity analysis which cannot consider the combined effects of random variables. Therefore, this paper aims to employ two robust methodologies of sensitivity analysis in order to obtain more reliable results. As a result, the main motivations of this study can be summarized as follows: (1) conduct sensitivity analysis of jacket type platforms against extreme wave loads to distinct the most important parameters affecting the nonlinear response of jacket platforms, (2) consider PSSI in the 3-D model of jacket, (3) utilize two robust methods of sensitivity analysis including Tornado and correlation coefficient approaches, (4) employ Incremental Wave Analysis (IWA) [12] as a newly introduced method in obtaining accurate behavior of jacket platforms against wave loading hazard and finally (6) consider both dynamic and static behavior of jacket platforms. 2. Modeling In this research, a 3-D model of SPD2 jacket platform located in South Pars Gas Field of Persian Gulf region is employed as a case study. General configuration of SPD2 jacket platform is displayed in Fig. 1. Since the same design specifications and physical configuration are applied to the offshore platforms in other South Pars Gas Field platforms, the results of this research can be valid for jacket structures in this area of Persian Gulf. SPD2 jacket platform located in 65 m water depth consists of six legs and three battered faces. Jacket plan dimension is about 16.00m × 27.50 m at deck level and 23.42m × 37.74m at mud line elevation. The jacket is fixed to the ground by 6 through leg grouted piles. The 3-D model of SPD2 jacket is modeled in accordance with AS-BUILT drawings. Furthermore, boat landing, risers, caissons, conductors and conductor guides, launch truss and barge bumpers are modeled as non-structural members. A simplified elastic model of topside including main framing is assumed. All topside loadings which are about 2020 ton are applied on main joints as equivalent point loads. The natural period of SPD2 jacket platform is about 1.5 s. The design wave height and period are also 12.2 m and 11 s, respectively. The finite element program USFOS, which has the capability to perform nonlinear static and dynamic analyses of jacket platforms against wave loadings, is selected. This software is proficient in taking into account the buckling and post buckling behavior of compression members as well as nonlinear PSSI, properly [13]. As nonlinear static and dynamic analyses against wave loading hazard are carried out in this research, both static and dynamic PSSI have to be considered. Static PSSI can be employed by means of static p–y, t–z and q–z curves described in API guidelines [14]. The vertical resistance of soil layers is modeled by means of two load-deformation curves, t–z and q–z, while lateral soil resistance is employed by nonlinear p–y curves. In this research, dynamic PSSI is considered utilizing “dynamic p–y” introduced by El Naggar and Bentley [15]. They

Fig. 1. General configuration of SPD2 platform located in Persian Gulf.

employed the static p–y curve approach in conjunction with the plane strain assumptions in order to represent the soil resistance within the frame of a Winkler model. They introduced a nonlinear

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spring model named “dynamic p–y”. The dynamic p–y curves are a function of the static p–y curve and velocity of the soil particles at a given depth and frequency of loading. The dynamic p–y curves are calculated using the following equation:
ω yn i h 2 P d ¼ P s α þ β ao þ κ ao d

ð1Þ

where Pd is the dynamic soil reaction at depth x (N/m), Ps is the static soil reaction (obtained from the static p–y curve) at depth x (N/m), ao is the dimensionless frequency = ω ro/Vs; ω is the frequency of loading (rad/s), ro is the pile radius (m), d is the pile diameter (m), y is the lateral pile deflection at depth x (m), when soil and pile are in contact during loading; and α, β, κ, and n are constants that depend on the soil type [15]. Despite the fact that collapse condition is considered in this research, failure modes of SPD2 jacket platform are described briefly. There are two main failure modes in SPD2 jacket platform comprising: (1) buckling of horizontal and diagonal bracing and (2) failure in soil and pile. While mean values of random variables are employed, buckling of bracing members happen immediately after failure in soil and pile. In other words, the SPD2 jacket platform has been designed in a way that the two failure modes happen approximately simultaneous. As a result, utilizing different values for random variables during sensitivity analyses can alter the order of failure modes. For example, when soil parameters are increased, the first failure mode (failure in bracing members) happens before failure in soil and pile.

149

curve for the SPD2 platform. Consequently, the point on cited curve, where sudden change in slope (e.g. less than 15% of the initial slope) occurs, will be defined as the Collapse Prevention (CP) limit state of the platform. The base shear (or overturning moment) corresponding to this point is considered as the ultimate capacity of platform. In addition, the wave height corresponding to the collapse prevention limit state is named Collapse Wave Height (CWH). The main advantages of this indicator as well as more details on the IWA approach and its applications in deterministic and probabilistic assessment of jacket platforms can be found in [12,16]. The main concepts of Dynamic Incremental Wave Analysis (DIWA) originate directly from Incremental Dynamic Analysis (IDA), which has been widely used in seismic evaluation of structures. To conduct DIWA, the structure is exposed to different incremental wave heights. Nonlinear dynamic analysis is carried out for individual regular wave heights, and the maximum structural demands (base shear, overturning moment, displacement, drift) are achieved accordingly. To gain DIWA capacity curve, the maximum base shear (or overturning moment) versus maximum displacement (or drift) extracted from each individual dynamic analysis is plotted. In DIWA procedure, the jacket structure is subjected to regular waves from Stokes' 5th order wave theory. The period of the regular wave is considered to be constant within each level of wave height intensity. Thus, the distribution of the wave period for each level of wave height is neglected in this research.

4. Uncertainties and random variables 3. Incremental Wave Analysis (IWA) In this paper, Incremental Wave Analysis (IWA) introduced by Golafshani et al. is employed as an advanced approach in obtaining the platform's response against wave loading [12]. IWA is able to estimate the ultimate capacity of jacket platforms against wave loading as well as displacement corresponding to the collapse point and a novel indicator named Collapse Wave Height (CWH). The benefits of this method have been comprehensively described in [12,16]. To conduct Static Incremental Wave Analysis (SIWA), the structural model should be subjected individually to incremental wave heights. For each individual wave height, nonlinear static analysis is carried out and the structural demand parameters (base shear, overturning moment, displacements, and drifts) are obtained accordingly. By performing this procedure, there would be finally a particular wave height at which the platform cannot undergo the wave loading, and the incremental analysis should be terminated. Plotting the base shear (or overturning moment) versus deck displacement (or drift), extracted from each individual analysis; will bring about a specific curve called IWA capacity curve. Fig. 2 illustrates the IWA capacity 40 CP limit State 35

Base Shear (MN)

Collapse Wave Height

Ultimate Capacity

30 25 20 15 10

Collapse Displacement

5 0

0

0.2

0.4

0.6

0.8

1

1.2

Deck Displacement (m) Fig. 2. IWA capacity curve for SPD2 platform.

1.4

1.6

The uncertainty and variability considered in the sensitivity analysis of this research are broken into three main categories including uncertainties in the prediction of the wave force on jacket structure, uncertainties in structural model and finally variability in properties of different soil layers (uncertainty in PSSI). The first category of uncertainty contains the main parameters influencing the wave force on jacket structure, which can be summarized as drag and inertia coefficients (Cd and Cm) as well as marine growth (MG). The uncertainties in the structural model account for the variability in the element and system levels of the jacket structure for a given design realization. The structural parameters which are considered to be random are yield stress of jacket piles and legs, fy,L, yield stress of jacket horizontal and diagonal braces, fy,B, yield stress of both groups including pile and legs and bracing members simultaneously, Fy, and modulus of elasticity, Es. In addition, vertical loads and masses are considered as system-level variables [16]. The third category of uncertainty in jacket type platforms is variability in characteristics of different soil layers. The soil profile at the location of the SPD2 platform similar to the majority places in Persian Gulf consists of a clay deposit with a total thickness of 120 m overlying bedrock. Therefore, the main properties of clay layers which affect the characteristics of nonlinear p–y, t–z and q–z springs should be taken into account as random variables. In this research, undrained shear strength of soil, Cu, unit weight, γ, and half of failure strain, ε50, are selected to examine the effects of uncertainty in soil properties and PSSI on the response of jacket platforms located in cohesive soils such as Persian Gulf region. Table 1 illustrates the mean or median values of random variables (based on the type of distributions) corresponding to the best estimates employed in the deterministic model as well as the coefficient of variation (COV) for each of the basic input random variables employed in the sensitivity analysis. Two different values for Cd, Cm, and MG indicate the above and below of splash zone, respectively. In addition, two different values for yield stress of steel material are illustrated based on the thickness of jacket members, which are in accordance with the design specifications of the case study jacket platform [16].

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Table 1 The statistical characteristics of random variables. Random variable

Mean or median

COV

Type

Reference

Parameters influencing variability of the wave force on jacket structure Drag coefficient Cd Inertia coefficient Cm Marine growth MG

0.65, 1.10 1.60, 1.27 75 mm, 50 mm

0.25 0.10 0.50

Lognormal Lognormal Lognormal

[17,18] [17,18] [17,18]

Parameters influencing modeling uncertainty Vertical loads and masses fy for pile and legs fy for braces fy for piles, legs and braces Modulus of elasticity

m and W Fy,L Fy,B Fy ES

Computed 335 MPa, 345 MPa 335 MPa, 345 MPa 335 MPa, 345 MPa 2.0601 × 105 MPa

0.10 0.07 0.07 0.07 0.03

Normal Lognormal Lognormal Lognormal Lognormal

[17,18] [17,18] [17,18] [17,18] [18,19]

Parameters influencing soil properties Undrained shear strength Unit weight Half of failure strain clay

Cu γ ε50

a

0.3 0.1 0.4

Normal Normal Normal

[20,21] [20,21] [20,21]

a

Symbol

a a

Deterministic values measured in each layer.

5. Sensitivity analysis utilizing Tornado approach An important task in the structural reliability analysis is to determine the significance of random variables, and how they influence the response of structure. In this way the dominating random variables associated with the considered limit state can be distinguished using certain sensitivity measures. In this study, sensitivity analysis is conducted by means of nonlinear pushover analysis as well as more advanced methods of SIWA and DIWA. Ultimate capacity and Reserve Strength Ratio (RSR) which are the main outcomes of pushover method are utilized as demand parameters in sensitivity analysis. In contrast, CWH, collapse displacement and finally a new representation of ultimate capacity are the chief indicators employed in the sensitivity analyses resulted from SIWA and DIWA. To determine the prominent parameters affecting the behavior of jacket platform, random variables listed in Table 1 are taken into account. random variable, two extreme values including pffiffiffi For each p ffiffiffi μ− 3σ and μ þ 3σ are taken, where μ and σ denote the mean/ median and standard deviation of each random variable. Then, the response of platform with regard to each random variable is attained, individually. This procedure is repeated for all random variables. Finally, a tornado diagram can be used to show the sensitivity of considered criterion to each random variable, graphically [22]. In

the following the results of sensitivity study for current pushover method, SIWA and DIWA are presented. It should be noted that variables are ordered by their relative importance.

5.1. Results of sensitivity analysis based on pushover practice utilizing Tornado approach Figs. 3 and 4 illustrate the tornado diagrams for ultimate capacity and Reserve strength Ratio (RSR) associated with the SPD2 platform. The RSR is defined as the ratio of platforms ultimate lateral load carrying capacity to its 100-year environmental condition lateral loading. From Fig. 3 it is seen that ultimate capacity in current pushover practice is only sensitive to undrained shear strength of clay soil layers and yield stress of members specially bracing elements. This matter dictates that the ultimate capacity of jacket platforms obtained by pushover practice is mainly related to random variables affecting the stiffness and strength of structure. Since drag and inertia coefficients as well as marine growth only alter the lateral load pattern of pushover analysis, they are not able to make a severe variation in the structural capacity. Fig. 3 also dictates that the lateral bracing system in jacket platforms is more influential in ultimate capacity of jacket platforms in comparison with framing members (legs). As a result, yield stress of leg members can be eliminated from reliability analysis of jacket structures as to reduce the number of random variables.

Ultimate Capacity (Pushover) RSR (Pushover) CU Fy

Cd

fy,B 10

CU MG

Mass

10

Fy

Es

fy,B

fy,L

Mass

Cm

Es

Cd

fy,L

Gama

Gama

E50

E50

MG

Cm 0

25

30

35

40

Ultimate Capacity (MN) Fig. 3. Tornado diagram showing the sensitivity of ultimate capacity to each random variable (pushover analysis).

0

2

3

4

5

RSR Fig. 4. Tornado diagram showing the sensitivity of RSR to each random variable (pushover analysis).

M. Hezarjaribi et al. / Journal of Constructional Steel Research 83 (2013) 147–155

Fig. 4 illustrates relative importance of different random variables regarding RSR parameter. The influential random variables can be broken into two main categories: (1) random variable affecting the ultimate capacity shown in Fig. 3 including undrained shear strength of clay soil layers and yield stress of members especially bracing members and (2) random variable altering the wave force on jacket structure (100 year wave force) including drag coefficient and marine growth. Despite the fact that RSR is defined as the ratio of a platform's ultimate capacity to its 100-year environmental condition loading, these random variables which affect the ultimate capacity and 100 year wave force are the most important parameters with regard to estimating of RSR indicator. The less importance of inertia coefficient than drag coefficient can be related to Morison equation used in calculation of wave force in which the drag coefficient is more dominant than inertia coefficient.

(a)

151

SIWA Sensitivity Analysis (Ultimate Capacity)

CU Gama E50

10

Fy Mass fy,L fy,B Cm MG Cd Es

5.2. Results of sensitivity analysis based on SIWA and DIWA utilizing Tornado approach

0

22

Figs. 5 to 7 show the sensitivity of SPD2 platform's response obtained by SIWA and DIWA to different random variables. Sensitivity of collapse displacement, ultimate capacity and CWH resulted from

28

31

34

Ultimate Capacity (MN)

(b)

DIWA Sensitivity Analysis (Ultimate Capacity)

CU

(a)

SIWA Sensitivity Analysis (Displacement)

fy,L

10

Mass

CU

fy,B

Fy Cm

25

10

Fy

Gama

Gama

E50

Cd

fy,B Cd

E50

Mass

Cm

Es

MG

fy,L

Es 0

MG

25 0.55

0.65

0.75

37

41

0.85 Fig. 6. Tornado diagrams showing the sensitivity of ultimate capacity of SPD2 platform to each random variable — (a) SIWA, (b) DIWA.

Displacement (m)

(b)

33

Ultimate Capacity (MN)

0

0.45

29

DIWA Sensitivity Analysis (Displacement)

both SIWA and DIWA can be interpreted using these figures. Observations from these figures are as follows:

CU fy,L fy,B Es MG Mass Cd E50 Gama Fy C,m 0

0.5

0.6

0.7

0.8

0.9

Displacement (m) Fig. 5. Tornado diagrams showing the sensitivity of collapse displacement of SPD2 platform to each random variable — (a) SIWA, (b) DIWA.

➢ Generally, soil properties especially undrained shear strength of clay in different soil layers mostly influence the behavior of jacket type platforms located in Persian Gulf region. This can easily be seen from tornado diagrams for all three indicators considered in this research which are ultimate capacity, CWH and collapse displacement. This matter directly stems from the lack of knowledge on the estimation of soil properties. It should be noted that the coefficient of variation of Cu is equal to 0.3 which is relatively high in comparison with other random variables. In addition, as mentioned before, the main failure mode of SPD2 jacket platform is failure in soil and pile. Despite the fact that variation in soil properties can alter the stiffness of system, its effects on the structural behavior of SPD2 are noticeable. Moreover, the undrained shear strength of clay is the most significant part of P–Y formula which can influence directly on the stiffness of P–Y springs. It is notable that both static and dynamic behaviors of jacket type platforms are affected by this category of uncertainty. This highlights the importance of PSSI on the behavior of jacket type platforms which coincides

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(a)

SIWA Sensitivity Analysis (CWH)

Cd CU E50 10 MG Gama Fy

➢

fy,B Mass fy,L

➢

Cm Es 0

17

19

21

23

Collapse Wave Height (m)

(b)

➢

DIWA Sensitivity Analysis (CWH)

Cd CU

➢

10

MG Mass

which can cause failure in the platform. As a result, it is clear that the CWH is directly related to amount of wave force applied to jacket platform. As drag coefficient is an essential part of Morrison equation, its variation can easily create fundamental changes in wave force and consequently change in the CWH. Marine growth parameter is also acting in the same way. It should be noted that marine growth can increase the hydraulic diameter of elements subjected to the waves which, in turn enhances the amount of wave force acting on jacket platforms. Ultimate capacity of jacket platform estimated by IWA approach is generally sensitive to soil properties and yield stress of jacket members. These random variables directly influence the strength and failure modes of SPD2 jacket platform. Collapse displacement is somewhat different between SIWA and DIWA results. Undrained shear strength of soil mostly influences the collapse displacement. As this random variable can alter the stiffness of structure considerably, its variations easily change the collapse displacement. Other random variables such as yield stress of members as well as module of elasticity have similar effects utilizing DIWA approach. Yield stress of structural elements is a significant parameter which affects the static and dynamic behaviors of SPD2 jacket platform. Its influence on different indicators can clearly be realized from the figures. It is seen that separating yield stress of legs from diagonal and horizontal members leads to different structural behaviors. The total load and mass do not play an important role in the static and dynamic behaviors of jacket type platforms.

5.3. First-order second-moment method

fy,L fy,B Gama E50 Cm Es Fy 0

17

19

21

23

Collapse Wave Height (m) Fig. 7. Tornado diagrams showing the sensitivity of collapse wave height of SPD2 platform to each random variable — (a) SIWA, (b) DIWA.

with the results obtained by other references such as [4,9,10]. ➢ With regard to CWH indicator, drag coefficient, undrained shear strength of soil layers and marine growth are the most significant random variables. As a result, two main groups influence the CWH of jacket platforms including parameters changing the wave force (Cd and MG) and parameters changing the strength of structure (soil properties). The main reason of this relation can be found in the basic concept of CWH indicator. CWH is a specific parameter which was first introduced in 2011 [12]. This indicator which is a significant conclusion of IWA approach estimates the ultimate capacity of structure utilizing an observable parameter which is in term of wave height. For example, the ultimate capacity of SPD2 platform utilizing mean values of random variables is a storm wave with 21 m height (CWH=21 m). As a result, it is clear that this indicator will be sensitive to the random variables changing the wave force and also the strength of structure. If it is assumed that the ultimate capacity of jacket platform is a constant value depending on the structural parameters such as Fy, Cu and so on, when the IWA approach is employed, the collapse prevention limit state happens in the least wave height which can produce wave loading equal to ultimate capacity and this wave height is called the Collapse Wave Height (CWH). In other words, the CWH is the smallest wave height

The previous sub-section discussed the sensitivity of the different indicators in jacket platforms including ultimate capacity, CWH and collapse displacement to selected random variables. This section takes this information and uses the first-order second-moment (FOSM) approximation to estimate the total uncertainty in the mentioned indicators that is caused by the uncertainty in the random variables. Baker and Cornell [23] have proposed this approach for use in performance-based earthquake-engineering studies. This method has also been employed previously by Ibarra [24] to estimate the variance of collapse capacity for single-degree-of-freedom systems. In the FOSM method, the target function, g, is represented as the first-order Taylor series expansion at the mean value point assuming that the variables X are statistically independent. In other words, the FOSM assumes that each random variable linearly affects the target function. This allows us to estimate the standard deviation of target function by different uncertainties considered in the research. However, the FOSM method is not able to evaluate how the mean (or median) of target function may be affected by considered uncertainties [25]. To begin the FOSM method, the target function, g, which relates the different random variables to examined indicators, for instance ultimate capacity, is written as: F ¼ g ðX 1 ; X 2 ; …; X n Þ

ð2Þ

where X1, X2, …, Xn are random variables, and F denotes the examined indicator such as ultimate capacity, CWH, etc. in order to determine the mean and standard deviation of F, the following equation can be employed [25]: μ F ¼ g ðMx Þ 2 " # n X n X ∂g ðX Þ ∂g ðX Þ 4 · σ ½F  ¼ ∂xi ∂xj i¼1 j¼1

ð3Þ 3 ρij σ i σ j 5

ð4Þ

x¼Mx

where Mx is the vector of mean values of the random variables, ∂g∂xðX Þ is i the gradient of the F with respect to random variable i, ρij is the

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correlation coefficient between two random variables i and j. Finally, σi is the standard deviation of random variable i. Despite the fact that no correlation is taken into account between different random variables, the Eq. (4) is rewritten as: 2  n  X ∂g ðX Þ σ ½F  ¼ 4 ∂xi i¼1

3 σ i 5:

ð5Þ

x¼Mx

153

Thus, above equation as well as results of sensitivity analysis obtained in previous section are employed in order to determine the standard deviation of response in SPD2 jacket platform. Table 2 summarizes the results of FOSM approach containing mean and standard deviations for CWH, ultimate capacity and collapse displacement. These values will be compared with the more exact results which are gained by more advanced solution of correlation coefficient approach in Section 6.

6. Sensitivity analysis utilizing correlation coefficient approach In this section, sensitivity analysis of SPD2 jacket platform to the aforementioned random variables is examined by a novel and advanced procedure introduced by Vorechovsky and Novak [26]. This method utilizes the statistical correlation between basic random variables and the structural response variable: a straightforward and simple approach. This approach utilizes the Latin Hypercube Sampling (LHS) in conjunction with Simulated Annealing (SA) optimization technique. LHS is a special type of Monte Carlo simulation which uses the stratification of the theoretical probability distribution functions of input random variables. The basic feature of LHS is that the probability distribution functions for all random variables are divided into Nsim equivalent intervals (Nsim is a number of simulations), and the values from the intervals are then used in the simulation process. The samples are chosen directly from the distribution function based on an inverse transformation of distribution function. In spite of high efficiency of LHS method, there are generally two problems concerning statistical correlation [26]: (1) diminishing undesired and spurious correlation between random variables generated during sampling procedure, particularly in the case of a very small number of simulations (tens) and (2) introducing the prescribed statistical correlations between pairs of random variables defined by the target correlation matrix. Hence, in order to overcome the above difficulties a novel stochastic optimization approach called Simulated Annealing (SA) is employed. Details of SA technique can be found in [26], while the practical application of this approach in reliability analysis of jacket platforms has been shown in [16]. To conduct sampling procedure according to LHS with SA technique, a computer code was prepared. Results of this code dictate that fifteen simulations (Nsim = 15) can give rise to the target correlation with appropriate accuracy for ten random variables indicated in Table 1. In order to demonstrate the efficiency of this algorithm, the target correlation matrix T is presented in the lower triangle of the following matrix, while the upper triangle illustrates the generated (actual) correlation A, 2 6 Cd 6 6 Cm 6 6 MG 6 6 F y;L 6 T¼6 6 F y;B 6 Es 6 6 Mass 6 6 C 6 u 4 γ ε50

Cd 1 0 0 0 0 0 0 0 0 0

Cm MG F y;L −0:0003 0:018 0:003 1 0:001 0:007 0 1 −0:014 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

F y;B 0:006 −0:007 −0:003 0:0186 1 0 0 0 0 0

Es 0:011 0:012 0:004 −0:011 −0:019 1 0 0 0 0

Mass 0:004 −0:008 −0:018 −0:007 0:0002 0:009 1 0 0 0

3 Cu γ ε50 −0:001 −0:011 0:0201 7 7 −0:005 0:0028 0:008 7 7 0:011 0:0198 0:012 7 7 0:0001 −0:007 0:012 7 7 −0:016 −0:0002 0:01 7 7: 0:0283 −0:0009 −0:012 7 7 0:0106 0:0005 −0:014 7 7 1 0:0039 −0:012 7 7 0 1 −0:013 5 0 0 1

It is noteworthy that LHS method in conjunction with SA technique leads to appropriate results, while the number of simulations is small in comparison with crude Monte Carlo simulation. To conduct sensitivity analysis for jacket type platforms using correlation coefficient method, first of all the sampling procedure is carried out by LHS and SA technique. Consequently, the structural response (different indicators such as ultimate capacity, CWH and collapse displacement) is achieved utilizing nonlinear static and dynamic incremental wave analyses (SIWA and DIWA) employing sampling values obtained in previous step. This sensitivity method utilizes the statistical correlation between basic random variables (sampling values for each random variable) and the structural response variable. In other words, the relative effect of each basic variable on the structural response can be measured using the partial correlation coefficient between each basic input variable and the response variable. The method is based on the assumption that the random variable which influences the response variable most considerably (either in a positive or negative sense) will have a higher absolute value of the correlation coefficient between sampled input and resulting output than other variables. In the case of a very weak influence the correlation coefficient will be quite close to zero [27]. Therefore, to conduct sensitivity analysis for the SPD2 jacket platform, more than 300 nonlinear static and dynamic analyses are carried out. Results of this approach are summarized in Tables 3 and 4 for SIWA and DIWA, respectively. The random variables have been arranged based on

Table 2 Results of FOSM for SPD2 jacket platform.

μ COV

SIWA DIWA SIWA DIWA

CWH (m)

Collapse displacement (m)

Ultimate capacity (MN)

20.50 21.00 0.18 0.16

0.69 0.73 0.32 0.29

33.2 37.9 0.25 0.22

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Table 3 Sensitivity factor of random variables resulted from correlation coefficient approach (SIWA). CWH Collapse displacement Ultimate capacity

Random variable Sensitivity factor Random variable Sensitivity factor Random variable Sensitivity factor

Cd −0.87 CU 0.62 CU 0.91

CU 0.44 E50 0.40 Fy,B 0.23

MG −0.28 Fy,B 0.24 Fy,L 0.17

Fy,B 0.09 Es −0.23 E50 0.13

Fy,L 0.08 Cd −0.21 MG 0.13

Es −0.03 Fy,L 0.21 Cm −0.09

Gama 0.03 MG 0.19 Cd 0.09

E50 0.02 Gama −0.16 Es 0.03

Mass −0.001 Cm −0.13 Gama 0.03

Cm −0.001 Mass −0.02 Mass 0.02

their importance on each indicator. These tables illustrate the correlation coefficient between output results and input variables for three different demand parameters considered in this study. The statistical correlation between basic random variables (sampling values for each random variable) and the considered indicator (CWH, collapse displacement and ultimate capacity) is a non-dimensional parameter which varies between 1 and − 1. This statistical correlation which illustrates the sensitivity of each demand parameter to different random variables is called “Sensitivity Factor” in this research. As mentioned above, the random variable which influences the response variable most considerably (either in a positive or negative sense) will have a higher absolute value of the sensitivity factor than other variables, while sensitivity factor close to zero dictates a very weak influence on the response. The observations from these tables confirm to some extent the results obtained by Tornado method. Undrained shear strength of soil layers is nearly the most dominant parameter in the behavior of SPD2 platform. In addition, the significant role of drag coefficient and marine growth on the CWH is obvious from these tables. Yield stress of bracing members is also affecting the ultimate capacity of SPD2 jacket platform which is again in line with Tornado results. It should be noted that values listed in these tables assist the researchers for choosing important random variables for reliability and probabilistic analyses of jacket platforms. For example, when the CWH indicator is employed in reliability analysis for evaluating the failure probability, only Cd, Cu and MG need to be utilized which cause a noticeable reduction in number of necessary simulations. More investigations are necessary to determine a reference value for importance of random variables used in reliability analysis. For example, random variables whose sensitivity factor (value in Tables 3 and 4) is fewer than 0.2 can be removed from reliability analysis procedure. In addition, Table 5 shows the mean and standard deviation of different indicators achieved by LHS and SA sampling together with the results obtained by FOSM in previous sections. Since the latter method can take into account the combined effect of different random variables, the results of this method are more reliable than the results of FOSM. Table 5 dictates that the difference in mean values achieved by both methods for SIWA and DIWA is less than 2 and 6%, respectively. In contrast, the FOSM is predicting the higher value for standard deviation compared to LHS approach. This difference is really noticeable, e.g., the values obtained by FOSM are 3 times greater than the results of correlation coefficient approach. This indicates that the FOSM cannot estimate the standard deviation of nonlinear response of jacket platforms, properly. The FOSM does not lead to suitable standard deviation when the level of nonlinearity of problem increases. The FOSM method assumes that each random variable linearly affects the collapse capacity. This allows us to predict how the standard deviation of collapse capacity is increased by the different uncertainties. In cases where a random variable nonlinearly affects the collapse capacity, the uncertainties will affect the mean collapse capacity in addition to the standard deviation ofpcollapse ffiffiffi capacity; the FOSM method cannot capture this and may be a considerable approximation in these cases [19]. In addition, p utilizing μ  3σ may ffiffiffi be another reason for this problem. Despite the fact that some random variables such as Cu have high COV, employing μ  3σ brings about two extreme values which can alter the target indicators (CWH, collapse displacement and ultimate capacity) noticeably and lead to higher standard deviation in comparison with LHS technique. Therefore, for jacket platforms located in Persian Gulf region, the mean value of nonlinear response obtained by FOSM method can be employed in probabilistic design or assessment of jacket platforms, while the standard deviation shall be achieved by more reliable approaches such as LHS.

7. Conclusion In this paper, sensitivity analysis of jacket type platform against wave loading hazard was carried out through a case study jacket structure located in Persian Gulf region. It is remarkable that this sensitivity analysis was conducted for both static and dynamic behavior of jacket platforms utilizing pushover practice as well as more advanced methods of Static and Dynamic Incremental Wave Analyses (SIWA and DIWA). In addition, two separate methods of sensitivity analysis were employed, namely Tornado approach and more advanced approach of correlation coefficient technique. The results of sensitivity analysis illustrated that soil properties appear to be the most important source of uncertainty in nonlinear static and dynamic behaviors of jacket platforms. The second important factor was the

yield stress of bracing members. In addition, the results dictated that drag coefficient can affect nonlinear behavior of jacket platforms in some ways. With regard to CWH indicator, the most important random variables are drag coefficient, undrained shear strength of soil layers and marine growth. Ultimate capacity of SPD2 jacket platform estimated by IWA approach is generally sensitive to soil properties and yield stress of jacket members. Undrained shear strength of soil mostly influences the collapse displacement. Other random variables such as yield stress of members as well as module of elasticity have similar effects on the collapse displacement. In addition, FOSM was utilized in order to estimate the mean and standard deviation of response. Although the FOSM estimated the mean values of response properly and its difference with more exact method of LHS was less than 2 and 6% for SIWA and DIWA

Table 4 Sensitivity factor of random variables resulted from correlation coefficient approach (DIWA). CWH Collapse displacement Ultimate capacity

Random variable Sensitivity factor Random variable Sensitivity factor Random variable Sensitivity factor

Cd −0.90 CU 0.56 CU 0.86

CU 0.37 Es −0.50 Cd −0.34

MG −0.33 Cd −0.46 E50 −0.26

Cm 0.08 MG −0.24 Fy,B 0.20

Gama 0.06 Fy,L 0.18 MG −0.16

Es −0.05 Gama −0.16 Gama −0.11

Fy,B 0.05 E50 0.15 Fy,L 0.11

E50 −0.04 Fy,B 0.11 Mass −0.002

Fy,L 0.04 Cm 0.02 Es −0.001

Mass 0.03 Mass −0.01 Cm 0.001

M. Hezarjaribi et al. / Journal of Constructional Steel Research 83 (2013) 147–155 Table 5 Mean and standard deviation achieved by FOSM and correlation coefficient approach.

μ

FOSM Correlation coefficient

COV

FOSM Correlation coefficient

SIWA DIWA SIWA DIWA SIWA DIWA SIWA DIWA

CWH (m)

Collapse displacement (m)

Ultimate capacity (MN)

20.50 21.00 20.30 20.30 0.18 0.16 0.08 0.08

0.69 0.73 0.70 0.69 0.32 0.29 0.10 0.07

33.2 37.9 33.1 35.7 0.25 0.22 0.08 0.10

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