The effect of gravity on the dynamic characteristics and fatigue life assessment of offshore structures

Journal of Constructional Steel Research 118 (2016) 1–21

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Journal of Constructional Steel Research

The effect of gravity on the dynamic characteristics and fatigue life assessment of offshore structures Junbo Jia Aker Solutions, Norway Sandslimarka 251, Sandsli NO-5861, Norway

a r t i c l e

i n f o

Article history: Received 20 November 2014 Received in revised form 11 September 2015 Accepted 24 September 2015 Available online xxxx Keywords: Gravity effects P-Delta effects Stress “stiffening/softening” Nonlinear dynamics Nonlinear waves Non-Gaussian response

a b s t r a c t This paper intends to draw attention to the influence of gravity on the dynamic response and fatigue damage assessment of offshore structures. In traditional fatigue life calculation, the gravity loads of structures are assumed to only contribute to the mean stress of the structures. This paper examines the significance of P-Delta (P-Δ) effects and the stress “stiffening/softening” induced by gravity. This paper first explains the two aforesaid gravityinduced effects, with studies on their influence on both eigenperiod and fatigue life assessment. A modal analysis of a typical offshore structure with large degrees of freedom is followed, to identify the dynamic characteristics influenced by the gravity effects. It is discovered that gravity load can induce a tendency to cause additional compressive and tensile forces to coexist in various structural components, causing the eigenperiods to increase, decrease or even cross each other. Compared to the stress “stiffening/softening” effects, the P-Delta effects on tuning the structure's stiffness and eigenperiods are insignificant in mild sea states. Furthermore, based on the nonlinear dynamic response calculation and a type of efficient wave energy inputs, a procedure for calculating fatigue damage is adopted and a systematic investigation of fatigue calculation of the structure is performed. It is discovered that ignorance of the gravity loads can underestimate the fatigue damage by up to 24%. Finally, through a series of investigations, it is discovered that gravity can have significant effects on both the response statistics and frequency content of the structural responses. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction As modern engineering structures become lighter and more slender than ever, and are also subjected to more adverse dynamic loading as a challenge, this has caused an increase in vibration amplitude, moved the vibration frequencies of structures into bands that are more awkward to deal with, and reduced the elastic critical load of structures. Therefore, identifying dynamic characteristics and fatigue assessment have become increasingly important, and almost routine for maintaining the structures' integrity. Linearisation involved in structural analysis can lose important information concerning structural responses, and therefore increase uncertainty in the integrity assessment. Consequently, nonlinearities should be taken into account. In a broad definition, the nonlinearities in structural analysis are reflected in the loading (e.g. fluid–structure interaction), geometric (e.g. large deformation and membrane and so on), material (e.g. nonlinear plasticity) and boundary (contact) aspects. For offshore structures, since frequency domain analysis can hardly account for the non-linear load effects induced from drag forces in Morison's equation [1–6], the variation of water surface causing the

E-mail addresses: [email protected], [email protected].

http://dx.doi.org/10.1016/j.jcsr.2015.09.013 0143-974X/© 2015 Elsevier Ltd. All rights reserved.

intermittency of wave loading, the variation of buoyancy forces on components in the splash zone [7–10], or large structural deformations, and also because the power spectrum of the critical stress due to the dynamic wave loading may not be narrow banded, time history analysis is a preferred method of reducing the aforesaid uncertainties. This is also used in the current paper to calculate the stress response for fatigue assessment. In traditional linear spectrum fatigue calculation in frequency domain, the gravity loads of structures only contribute to the mean stress of the structures. This of course influences the ultimate limit state evaluations of offshore structures. However, due to the large deformation of the offshore structures subjected to waves, and the centres of gravity for structures also changing over time, the gravity loads of topside and its supporting structure exert additional actions on the structure. These additional actions vary with the variation of the structure's deflection, i.e. they also vary over time. The additional actions in turn change the stiffness of the structure; this is a type of P-Delta (P-Δ) effect [11]. P-Delta effects can play an important role in increasing the effective load on a structural bay. Ruge [12] first exams the P-Delta effects to estimate the change of deflection of a vertical cantilever beam supporting a weight. Jennings and Husid [13] first study the gravity effects on inelastic structural responses, in which they emphasise the significant contribution of the effect of gravity. Montgomery [14] states that the influence of P-

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J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

Delta effects are important if the ratio of base shear to frame weight is less than 0.1, or if displacement ductility demands are above 2.0. In seismic analysis, Gupta and Krawinkler [15] study 1 (2000), pp. 145–154. the seismic response of multiple degrees of freedom systems to earthquake excitation. They state that P-Delta effects can exert significant influence on dynamic response, particularly for multi-storey moment– resistant frames that can develop large inter-storey drifts. However, P-Delta effects are normally ignored by offshore engineers in their daily engineering practice. The main reason for this is that previous research focuses on P-Delta effects with regard to the ultimate limit state assessment, which indicates that the effects are relatively small, especially for offshore structures in shallow to medium depths [16]. Another important reason is that the nonlinearities due to the structure's large deformation cannot be properly handled in conventional linear spectrum analysis, and engineers normally take it for granted that the additional actions due to gravity loads are too small to be taken into account. Unlike evaluations of ultimate limit state, fatigue damage is proportional to stress amplitude, with power typically in the range from 3.0 to 5.0; a slight variation in stress amplitude may induce significant change to the fatigue damage calculations, and the nonlinear structural deflections due to gravity loads may then be a significant contributor to influencing fatigue damage. In addition, the stress “stiffening/softening” effects due to the gravity loads also slightly change the structure's natural frequencies, which can influence the fatigue damage of the structures as well. Particularly for structures under compressive forces globally, the natural frequency normally decreases, and this can cause problems regarding vibration and fatigue. The earliest publication to address the stress “stiffening/softening” under axial load was written by Galef [17], who set outs a practical formula for calculating the natural frequency under compressive load. Based on Galef's work, Bokaian [18] investigates the applicability of Galef's formula under a set of support conditions and also extends his study to tension loads [19]. Using Galef's formula, Shaker [20] first investigates relevant practical problems in aerospace structures: a vibrating beam with arbitrary boundary conditions, a cantilever beam with tip mass under constant axial loads and a cantilever beam with tip mass under axial loads applied on the tip directed to the root. In addition, he also extends his analysis to tension loads. Virgin [21], Virgin, Santillan, and Holland [22] and Virgin and Plaut [23] also perform a series of research to investigate the dynamic behaviour of axially-loaded structures. By neglecting the large deformation of vertical cantilevers, Virgin and his co-workers [22] calculate their eigenfrequencies and validate the calculation through a number of modal testings. They experimentally proved that a beam with an “upward” orientation will experience de-stiffening effects and a beam in a “downward” orientation will be stiffened by the weight of the beam through the development of tension stress in the beam. By including the effects of large displacement equilibrium paths through a nonlinear moment-curvature relationship, Virgin and Plaut [23] further present a formulation to obtain the eigenpairs of vertical cantilevers. From a mathematical perspective, buckling and vibration are both eigen-oriented: buckling occurs when compressive stresses are large enough to lead to a zero resultant stiffness condition, i.e. the natural frequency of the structure reaches zero [24]. By performing a nonlinear time domain dynamic analysis, and calculating the fatigue damage, Jia [25] investigated the dynamic responses and the resulted fatigue damage of a typical flare boom subject to dynamic wind loading, from the calculation it is found out that gravity load generally gives more deviation of the kurtosis from a value of three, i.e. the gravity load increases the non-Gaussian trend of the response. In general, for the welded joints under study, the calculated fatigue lives by accounting for the gravity effects are significantly less than the ones without considering the gravity effects [26]. Several pieces of research [27–29] discuss the use of measured vibration frequencies obtained from non-destructive modal testing to determine approximate buckling loads. They all conclude that changes in the measured vibration frequencies during increased loading can be used to predict the buckling of a structure. With the aim of

developing robust low-dimensional models, Mazzilli and his coworkers [30] use the nonlinear normal mode method to develop a rigorous derivation of non-linear equations that govern the dynamics of an axially-loaded beam; they also apply the equations for a study of dynamic characteristics of offshore risers. However, research into fatigue damage assessment influenced by gravity effects is limited. Through the calculation of wind buffetinginduced fatigue for a high-rise flare boom structure, Jia [31] states that influence due to gravity loads is significant on certain parts of the flare boom. To address the situation of limited research work in this area, this paper first explains the two aforesaid gravity effects. Furthermore, the dynamic characteristics and gravity effects of a realistic offshore structure are studied. By using different numbers of discrete frequencies, as well as formulating the frequency spacing with uneven distances as a type of efficient wave energy input, and by calculating the nonlinear structural dynamic response in time domain, a procedure presented by Jia [32] and Jia, Ellefsen and Holmås [33] for calculating the fatigue damage of offshore structures is adopted to carry out investigations into gravity effects. Furthermore, their influences are also studied through a series of statistical and frequency checks of the structure's response. 2. P-Delta and stress “stiffening/softening” effects Traditionally, the self-weight of a structure is considered to only cause a constant load (induces mean stresses) on the structure, which has almost nothing to do with the fatigue calculation of welded joints (not influenced by the mean stress due to the presence of residual stress). However, in reality two types of effect induced by self-weight are relevant: the P-Delta (P-Δ) effects and stress “stiffening/softening”. The P-Delta effects are due to the variation of a structure's lateral deformation in the horizontal plane, which changes the resultant action point of the structure's self-weight, and consequently induces additional actions on the structure and changes its stiffness. The change of stiffness further changes the force distribution on the structure. These effects are obviously nonlinear and can start a pernicious circle against the structural system because the influence of gravity loads increases as the lateral displacement grows, while at the same time the lateral displacement is magnified as a consequence of gravity loads acting on them [34]. For many moment-resistant frame structures, the P-Delta effects associated with component deformation are insignificant compared to the P-Delta effects on the displacements at the ends of the components [35]. In addition, the stiffness changes due to the P-Delta effects also alter the eigenfrequencies of the structure. However, this type of effect cannot be handled in conventional linear spectrum fatigue analysis. In this study, by adopting the implicit algorithm HHT-α method for the time integration in nonlinear dynamic analysis, both the stiffness and load matrix are updated at each time step — the large deformation-induced geometric nonlinearities can then be taken into account. The stress “stiffening/softening” derives from the fact that the selfweight may introduce significant component force globally. For offshore structures, this typically causes compressive force, which alone does not contribute to the calculated fatigue damage for welded joints. Meanwhile, since it decreases the global stiffness of the structure, the eigenfrequency is therefore decreased (stress “softening” or geometric “softening”), thus influencing the dynamic responses of structures, consequently affecting the fatigue damage accumulation as well. Eqs. (1) and (2) show the formulation of linearised buckling and vibration-generalised eigenproblem respectively. ð−KN Þφ ¼ λb KS φ

ð1Þ

KΨ ¼ λMΨ

ð2Þ

Where K is the linear stiffness matrix, KN is the nonlinear strain (geometric or initial stress) stiffness matrix, KS is the linear strain stiffness

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

3

matrix, and λb and φ are the eigenvalue and eigenvector for the linearized buckling solutions respectively. K and M are the stiffness matrix and mass matrix of the structure respectively. λ and Ψ are the eigenvalue and eigenvector for the vibration eigenproblem solutions respectively. It is known that the buckling eigenproblems are only related to the stiffness of the structure, while it is the dynamic eigenproblem that relates to both the stiffness and the mass distribution of structures. Note that both the modal analysis and buckling analysis are eigenorientated, meaning they have a close relation to each other — for example, the buckling takes place when, as a result of subtracting stress stiffness induced by compressive load from elastic stiffness, the resultant structure stiffness drops to zero; and the vibration modes occur when, as a result of subtracting inertia stiffness from elastic stiffness, the resultant stiffness is decreased to zero; the first natural frequency of a structure decreases with the increase of the compressive stress in the structure; when the frequency reaches zero, the corresponding load magnitude equals the buckling load [24]. For a prismatic beam, Galef's formula [17,18,36,37], expressed in Eq. (3), shows the relationship between the eigenfrequency fn under axial force P (P b 0 for compressive force and P N 0 for tension force) and the eigenfrequency of the uncompressed/untensioned beam fn,noP. f n ¼ f n;noP 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 1þ 2 n  Pe

ð3Þ

Where Pe is the Euler critical buckling load and n is order of the eigenfrequency. Fig. 1 shows a simplified physical modelling of a typical offshore platform structure, for which the Galef's formula can be applied by neglecting the distributed load q, therefore P =M ⋅g. The upper figure in Fig. 2 shows the eigenfrequency ratio due to the presence of the axial compressive force. It is noted that, due to the moderating influence of the factor n12 , the effect of the axial force is much more pronounced for the natural frequency (1st eigenfrequency) than the higher order eigenfrequencies. It should be noted that the support conditions of a beam limit the applicability of Galef's formula: for the fundamental eigenmode (n = 1 in Eq. (3)), Galef's formula is valid for pinned-pinned, sliding-pinned and sliding-sliding beams, approximate for sliding-free, clamped-free (the case for the current analysis), clamped-pinned, clamped-clamped and clamped-sliding beams, while not valid for pinned-free and free-free beams. However, for the third and higher modes of vibrations, Galef's formula can be applied to all types of support conditions [18,19,38]. Note that the fatigue damage accumulation experienced by a fixed offshore structure, such as a jacket or jack-up structure, is mainly caused by the inertia-dominated wave loadings due to small and medium sized waves. Therefore, by neglecting the wave-induced drag forces, and assuming that the fundamental eigenmode (n = 1) dominates the dynamic response of the structure, i.e. the quasi-static contribution to the mean square stress is assumed to be small, the ratio of fatigue damage at two different natural frequencies can be expressed by Eq. (4): FL0 ¼ FL



0

f1 f1



σ 0dl σ dl

m ð4Þ

where σdl and σdl′ are the mean square root stress from the dynamic response of the vibration mode at the two different natural frequencies. Depending on whether the source of natural frequency change is stiffness or mass, the mean square root stress is influenced differently 0. For stiffness change-dominated dynamic response, σdl can be expressed in Eq. (5):

σ dl ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 Sðω1 Þ : M 1 ω1 5

ð5Þ

Fig. 1. Visualisation of simplified offshore structure modelling: a vertical beam fixed at the bottom with a concentrated topside mass.

For mass change-dominated dynamic response, σdl can be expressed in Eq. (6):

σ dl

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 Sðω1 Þ ¼ K 1 ω1 3

ð6Þ

where A is a constant depending on structural and wave characteristics such as damping and wave spreading. M1 and K1 are modal stiffness and modal mass, and are expressed by Eq. (7): K 1 ¼ M 1  ω1 2 :

ð7Þ

For the wind-driven sea state, the upper boundary of the wave spectra value can be modelled at frequencies higher than the frequency of the peak [39], which is expressed in Eq. (8) S max ðωÞ ¼ 2:5783  10−4

 ω −4:6 2π

:

ð8Þ

By assuming that for small variations in natural frequency, the ratio between mean square modal deflection and mean square stress remains constant, i.e. the mode shape does not change with the variation of natural frequency 0, and also by assuming that all of the constants of the spectra in Eq. (8) are absorbed into the constant A, the Eqs. (5) and (6) can be rewritten as Eqs. (9) and (10) respectively:

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J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

For mass change-dominated cases:

σ dl

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 1 ¼ : K 1 ω1 7:6

ð10Þ

By substituting Eqs. (9) and (10) into Eq. (4), one finally obtains the ratio of fatigue damage as Eq. (11) for stiffness change-dominated cases and Eq. (12) for mass change-dominated cases: FL0 ¼ FL FL0 ¼ FL





0

f1 f1 0

f1 f1

ð−4:8mþ1Þ ð11Þ ð−3:8mþ1Þ ð12Þ

where m is the negative inverse slope of the S–N curve for the material. Based on Eqs. (11) and (12), the middle figure of Fig. 2 shows that the upper boundary of fatigue life ratio varies with natural frequency. It is noted that a 1% variation of natural frequency can result in a fatigue life change of up to 13%, 17%, and 21% for m = 3.0, m = 4.1 and m = 5.0 respectively. By matching the effects of gravity on natural frequency in the upper figure of Fig. 2 and the effects of natural frequency variation on fatigue life in the middle figure of Fig. 2, the upper boundary of fatigue life ratio varying with the axial compressive force is plotted in the lower figure of Fig. 2. Note that in a practical case, both quasi-static and dynamic responses contribute to mean square stress, while the derivation of Eqs. (11) and (12) omits the contribution from the quasi-static part of the responses. Therefore, the results from these equations are regarded as the upper boundary solutions. In reality, the sensitivity of fatigue life to the variation in natural frequency may not be so high. 3. Description of the adopted wave spectrum The wave environment comprises sea states, which are random processes described by random wave models using a wave spectrum. The model may be visualised as the summation of a large number of periodic wavelets, each of these wavelets having its own direction, amplitude and frequency [40]. Typical wave energy expression in a frequency domain takes the form expressed in Eq. (13): SPM ðωÞ ¼

A − D4 e ω ω5

ð13Þ

Fig. 2. The eigenfrequency ratio (upper figure) and the fatigue life ratio (middle and lower figure) due to variation of natural frequency presence of the axial compressive force P, Pe: Euler critical buckling load, fn and F: the eigenfrequency and fatigue life with the presence of the axial compressive force, fnoP: and FnoP: the eigenfrequency and fatigue life without the presence of the compressive force, m: negative inverse slope of the S–N curve for the material.

For stiffness change-dominated cases:

σ dl

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 1 ¼ : M1 ω1 9:6

ð9Þ

Fig. 3. Illustration of the JONSWAP spectra and the sampling frequency with the N = 64 samples (Δ) per rad.

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

5

Project can be adopted. Compared to the PM spectrum, the JONSWAP spectrum is narrow banded and extensively adopted by the offshore industry. It is expressed by enhancing the peak of the PM spectrum as shown in Eq. (16) SJONSWAP ðωÞ ¼

A − D4 δ e ω γ ω5

ð16Þ ðω−ωm Þ2 2 2

−

where A =a ⋅ g 2, D ¼ 1:25  ωm 2 , δ ¼ e 2σ ωm , g = 9.8 m/s2. a represents the level of high frequency tail, and is taken as 0.0081 in this study. g is the acceleration of gravity. The JONSWAP spectra have five free parameters: a, ωm, γ, σa, and σb. ωm is the peak angular frequency. The γ value indicates the enhancement of the spectrum peak, normally taken between 1 and 7. It is recommended as 2 by standards in offshore design such as the NORSOK Standard [41]. σ represents the narrowness of the peak, and has a different value for frequencies lower (σa) and higher (σb) than the peak frequency ωm as expressed in Eq. (17) [41]:  σ¼

σ a ¼ 0:07; ω b ωm σ b ¼ 0:09; ω N ωm

ð17Þ

By discretising the area below the wave energy spectrum into N number of components ΔS(ωsn) between ωn − Δω/2 and ωn + Δω/2 as shown in Fig. 3, the relationship between the wave energy spectrum and the amplitude of wave components can be approximated as Eq. (18): an ¼

Fig. 4. Geometry of the topside, jacket and the location of sea surface.

where A and D are two coefficients determined by wave characteristics, and ω is the angular frequency of the wave. For fully developed seas, the Pierson–Moskowits (PM) spectrum can be adopted to generate the wave elevation. This spectrum is used for the fully developed wave condition, i.e. the fetch and the duration are large and there is no disturbance from other areas. This is due to the fact that after a certain period of wind blowing, the sea elevation becomes statistically stable. In addition, it is mainly developed for the description of waves in the North At̅

lantic Ocean. It can be expressed using the significant wave height H 1=3 ̅

and zero crossing mean period T 0 as independent parameters and input into the typical wave energy expression (Eq. (13)), as expressed in Eqs. (14) and (15): A ¼ 123:95H 1=3 ̅

2

=T 0 ̅

4

ð14Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ΔSðωn Þ

ð18Þ

where n = 1, 2, …, N. Provided Δω is constant, the wave elevation will be repeated with a period of 2π/Δω. In order to increase this period, a large number of N of discrete frequencies, as well as giving the frequency spacing uneven distances, should be used for the energy inputs to calculate the wave loads. In this study, after setting up the number of discrete frequencies N, the uneven distance is decided based on the criteria that each number of components ΔS(ωn) has the same zero spectral moments (area under the energy spectrum for each number of components ΔS(ωn)). This implies that more sample values are chosen near the peak frequency region than far from the peak frequency range. This is generally preferred. However, the main drawbacks of this is that if the number of discrete frequencies N is not high, one may miss a very important contribution of the wave energy spectrum tail in the high modal wave period range. In this study, the window of the wave energy spectrum is chosen between the frequencies of 0.05 Hz to 0.33 Hz. And the Stokes 5th order wave theory is adopted to generate the wave elevations. The linearised (Airy) wave theory [42,43] is adopted to model wave elevation, water particle velocity and accelerations. The water surface elevation ξ(x,t) of an irregular wave at location x and time t can then be calculated by the linear superposition of the wave components an as expressed in Eq. (19): ξðx; t Þ ¼

N X

an cosðkn x−ωn t þ γn Þ

ð19Þ

n¼1

Table 1 Hydrodynamic drag and inertia coefficients used in the calculation. New NORSOK Standard reversion 2007 [52]

̅ 4

D ¼ 495:8=T 0 s:

ð15Þ

In shallow waters with limited fetch and for extreme wave conditions, the JONSWAP spectrum developed by the Joint North Sea Wave

CD CM

Conductor

Jacket structure

1.15 Above elevation +2 m: 1.6 Below elevation +2 m: 1.2

1.15 (1.265 if anode is present)

6

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

where: • • • •

an is the amplitude of component n; ωn is the angular frequency of component n; kn is the wave number; γn is the random wave phase that ranges from −π to π.

In the absence of current, the wave number is related to the wave frequency by ω2n ¼ g  kn tanhðkn  dÞ where d and g are the water depth and acceleration of gravity respectively. Associated with the water surface elevation in Eq. (19), at a position (x, z) and time t, with a water depth d, the horizontal and vertical water particle velocity u and w respectively can be expressed as Eqs. (20) and (21): uðx; z; t Þ ¼

X

an  ωn  χ hn  cosðkn x−ωn t þ γ n Þ

ð20Þ

n

wðx; z; t Þ ¼

X

an  ωn  χ vn  sinðkn x−ωn t þ γ n Þ

ð21Þ

n

where the horizontal and vertical depth attenuation χhn and χvn can be expressed as Eqs. (22) and (23):

Fig. 5. Wave directional probability.

χ hn ¼

cosh½kn ðz þ dÞ sinhðkn dÞ

ð22Þ

drag force effects [45], and also because fatigue damage accumulation is mainly due to the contribution of small and medium-sized waves, the nonlinearity of wave itself is therefore disregarded in this study.

χ vn ¼

sinh½kn ðz þ dÞ : sinhðkn dÞ

ð23Þ

4. Description of HHT-α method for time integration

By differentiating the velocity equations with respect to time t, the horizontal and vertical water particle accelerations can then be written as Eqs. (24) and (25): •

uðx; z; t Þ ¼

X

an  ω2n  χ hn  sinðkn x−ωn t þ γ n Þ

ð24Þ

n

X • an  ω2n  χ vn  cosðkn x−ωn t þ γn Þ: wðx; z; t Þ ¼ −

Numerically, dynamic analysis of structures with large degrees of freedom is mostly carried out in two steps. The first step is to discretise the equations in space using finite element method, and the second step is to use a decent time-stepping method to integrate the equations in time. A system of second-order differential equations can be used to express the dynamic equilibrium. This system of equations is expressed here in matrix form as Eqs. (26) and (27):

ð25Þ

n

••

•

½M fU g þ½C  fU g þ½K fU g ¼ fRðt Þg

According to linear random wave theory, the water surface elevations and particle kinematics (the velocity and the acceleration) are all Gaussian-distributed and are therefore fully defined by the variances of their associated probability distributions, which are equal to the area under their frequency spectra. It should be noted that, in severe sea states, water surface elevations and particle kinematics will not follow the Gaussian distributions [44]. Nevertheless, this nonlinear nonGaussian effect of wave itself is of less importance than the nonlinear

Table 2 The probability of 22 sea states used in the fatigue analysis. Block

Hs [m]

Tp [s]

Pb

Block

Hs [m]

Tp [s]

Pb

1 2 3 4 5 6 7 8 9 10 11

1.4 1.5 1.5 1.6 1.5 2.4 2.9 3.0 3.1 3.2 4.6

5.0 7.5 10.3 13.1 16.1 5.4 7.8 10.3 13.2 16.0 8.3

0.060 0.178 0.103 0.024 0.003 0.008 0.184 0.173 0.062 0.010 0.024

12 13 14 15 16 17 18 19 20 21 22

4.9 4.9 4.9 6.7 6.9 6.8 8.6 8.8 8.9 10.7 12.8

10.4 13.2 16.2 10.8 13.2 16.1 11.2 13.2 16.0 13.4 13.0

0.087 0.033 0.007 0.018 0.013 0.002 0.002 0.004 0.001 0.001 0.00004

ð26Þ

where: •

fU g ¼

  2 •• d d • d fU g and fU g ¼ U ¼ 2 fU g: dt dt dt

ð27Þ

In Eq. (7), {U} is a matrix of displacements; {R(t)} is a matrix of external forces applied to the system; [M] is the mass matrix; [C] is the damping matrix; and [K] is the stiffness matrix. The second order differential equations of motions can be solved mainly using one of two numerical strategies: one is direct integration method, and the other is modal superposition method. The main difference between these two strategies is that in the modal superposition

Table 3 Marine growth profile data used in the modelling. Elevation

Thickness (mm)

Above 2.1 m +2 m to +2.1 m −38.7 m to +2 m −40 m to −38.7 m Below −40 m

0 Linear decrease from 100 to 0 100 Linear increase from 20 to 100 20

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

7

Fig. 6. Analysis procedure for calculating wave-induced fatigue damage [54].

method, a transformation is always performed prior to numerical integration. The direct integration method uses a numerical step-by-step procedure for integration of the governing Eq. (26). The most commonly used methods are the central difference method, the Houbolt method, the Wilson-θ method, and the Newmark method [46]. In the USFOS computational programme used in this study, the HHT-α method is adopted to solve the second order differential equations. This method is a oneparameter, multi-step implicit method and it employs the time averaging of the damping, stiffness and load term expressed by the αparameter [47]. The advantage of this method is that it introduces the numerical damping of higher frequency modes without degrading

accuracy. The governing equilibrium equation can then be expressed as Eq. (28) with the assumptions defined in Eqs. (29) and (30): ••

•

•

½MfU gnþ1 þ ð1 þ α Þ½C fU gnþ1 −α ½C  fU gn þð1 þ α Þ ½K fU gnþ1 −α ½K fU gn ¼ ð1 þ α ÞfRgnþ1 −α fRgn

ð28Þ

where •

fU gnþ1 ¼ fU gn þ ΔtfU gn þ

•• •• γt n 2 ð1−2βÞU n þ Δt 2 βU nþ1 2

ð29Þ

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Table 4 First ten eigenperiods of the jacket-topside-flare tower structures (hydrodynamic added masses are included) for cases with gravity and without gravity effects, in calm sea conditions. Mode Eigenperiod (s) Comments number With Without Diff. gravity gravity (%) 1 2 3 4

4.173 4.115 2.453 1.203

4.120 4.065 2.442 1.194

1.3 1.2 0.4 0.7

5

1.193

1.187

0.5

NA

1.152

NA

6

1.090

1.086

0.4

7

1.014

NA

NA

8

0.990

0.987

0.3

9

0.914

0.939

2.7

10

0.902

0.901

0.1

•

•

••

The first global flexural vibration along the east–west direction (Y). The first global flexural vibration along the south–north direction (X). The first global torsional vibration mode along the vertical axis that goes through the centre of the jacket structure. A global flexural vibration along the east–west direction; the flare tower and topside show more significant flexural vibration than that of the jacket. A local vibration of the centraliser on the topside. A global flexural vibration along the east–west direction, the flare tower and topside show more significant flexural vibration than that of the jacket. A local vertical vibration at the bottom horizontal frames together with the global flexural vibration along the south–north direction, the flare tower and topside show more significant flexural vibration than that of the jacket. A global flexural vibration along the south–north direction, the flare tower and topside show more significant flexural vibration than that of the jacket. A local vertical vibration at the bottom horizontal frames together with the global flexural vibration along the south–north direction, the flare tower and topside show more significant flexural vibration than that of the jacket. A global flexural vibration along the south–north direction, the flare tower and topside show more significant flexural vibration than that of the jacket. The flare tower vibrations are out of phase with the vibration of the jacket-topside structure. A local vibration of the centraliser on the topside together with slight global flexural vibration along the south–north direction for the flare tower and the jacket. A global flexural vibration along the east–west direction, the flare tower and topside show more significant flexural vibration than that of the jacket. The flare tower vibrations are out of phase with the vibration of jacket-topside structures.

••

fU gnþ1 ¼ fU gn þ ΔtγfU gn þ ΔtγfU gnþ1

ð30Þ

where n is the order number of the time step; Δt = tn+1 − tn, is the time increment; β and γ are the coefficients that, together with α, determine the integration accuracy and stability. The coefficient β denotes the variation in acceleration during the time-incremental step. In the original Newmark-β method (α = 0), γ is set equal to 0.5 to avoid artificial damping. Different values of β indicate different schemes of interpolation of the acceleration over each time-step. For example, β = 0 indicates a scheme equivalent to the central difference method, and β = 1/6 corresponds to the linear acceleration method; the linear acceleration approach can also be obtained by setting θ = 1 in the Wilson-θ method — see reference [46]. In the HHT-α method, unconditional

stability is achieved by satisfying the following conditions as expressed in Eqs. (31)–(33). The default α value adopted in USFOS is 0. −

1 bαb0 3

ð31Þ

β¼

1 ð1−α Þ2 4

ð32Þ

γ¼

1 ð1−2α Þ 2

ð33Þ

Fig. 7. The global flexural (1st), torsional (3rd) and significant flare tower flexural (4th) vibration eigenmodes.

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

9

is likely to be a full matrix, which requires higher calculation costs. To limit the bandwidth of the stiffness matrix, this expansion reduces to the Rayleigh-damping form when the series is truncated after the two first terms, as adopted in this study. The Rayleigh damping factors are calculated by the solution of a pair of equations with the damping ratio associated with two specific modes defined by the user. Compared to the explicit method, the current implicit HHT-α integration method is numerically stable for any Δt. The time step selection is then only based on accuracy considerations. However, the calculation for each time step is more costly for implicit method than that of the explicit method, and the number of operations in the direct integration method is directly proportional to the number of time-steps in the analysis. Hence, the chosen Δtshould be as large as possible without losing the required accuracy.

Fig. 8. The global base shear, overturning moment and horizontal displacement at topside CoG due to the platform south (upper figure) and west (lower) wave with Hs = 12.8 m, and Tp = 13 s.

For computational reasons, the damping matrix is expressed in terms of Caughey series as written in Eq. (34). ½C  ¼

X

 k α k ½M  ½M−1 ½K 

ð34Þ

k

The weight factors αk are calculated from modal damping of structures [47]. Note that if k is larger than 2, the damping matrix

Table 5 First three eigenperiods of the jacket-topside-flare tower structures (hydrodynamic added masses are included) for cases with gravity and without gravity effects, wave from platform south or platform west, Hs = 12.8 m and Tp = 13 s. Mode number

Wave platform direction

Eigenperiod (s) With gravity 2 s

1 2 3

Wave from south Wave from west Wave from south Wave from west Wave from south Wave from west

4 s

13.75 s

Without gravity 19.25 s

2 s

4 s

13.75 s

19.25 s

4.121 4.173 4.066

4.065

4.115 2.442 2.453

Fig. 9. The selected positions for fatigue calculation.

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5. Analysis model

Table 6 Calculated fatigue life at the selected joints on the jacket (the number of discrete sample frequencies in the JONSWAP wave spectrum: N = 120).

5.1. Structural prototype and modelling Most of the steel offshore supporting structures are threedimensional frames fabricated from tubular steel components. This gives the best compromise for satisfying the requirements of low drag coefficients, high buoyancy and high strength to weight ratio [48]. The most commonly used offshore structure is a jacket structure, which comprises a prefabricated steel support structure (jacket) extended from the seabed (connected with piles at seabed) to some height above the water surface level, and a steel deck (topside) on the top of the jacket. Because the use of tubular components gives rise to significantly high stress concentrations in the joints, fatigue life is one of the major concerns for the tubular offshore structures. For example, it is reported by Stacey and Sharp [49] that fatigue cracking has been a principal cause of damage to North Sea structures and in-service experience has shown that there have been several incidents of fatigue cracking requiring repair. In some cases, fatigue cracking has led to components severing, resulting in a consequent reduction in overall structural integrity. The prototype jacket structure considered in this study is a typical jacket structure comprising both the topside and the jacket with four legs, as shown in Fig. 4. The height of the jacket is 181 m with a water depth of 157 m. The weight for the topside and jacket is 12,904 tonnes and 7100 tonnes respectively. In total, 1789 two-node beam elements were used to model both the topside and the jacket. The structure is modelled as steel material with Young's modulus of 210 GPa, Poisson's ratio of 0.3, and a density of 7850 kg/m3. Lumped masses are modelled on the topside to represent the weight of the equipment and other nonstructural installations. For fatigue analysis, NORSOK Standard N-004 [50] requires that structure-to-ground connections may normally be simulated by linear stiffness matrices. And the matrices must account for the correlation between the rotational and translational degrees of freedom; this may be important for proper calculation of fatigue lives in the lower part of the structure. Linear springs at seabed are modelled to simulate both the pile-soil stiffness and the conductor connection with the seabed. The force deformation relationships (stiffness) for them are expressed in Eqs. (35) and (36) respectively: 3 2 1825 Fx 6 Fy 7 6 0 7 6 6 6 Fz 7 6 0 7 6 6 6 Mx 7 ¼ 6 0 7 6 6 4 My 5 4 0 0 Mz 2

0 49 0 0 0 −550

3 2 665 0 Fx 6 Fy 7 6 0 7:1 7 6 6 6 Fz 7 6 0 0 7 6 6 6 Mx 7 ¼ 6 0 0 7 6 6 4 My 5 4 0 0 0 −31:4 Mz 2

0 0 0 0 0 0 49 0 550 0 4079 0 550 0 10; 000 0 0 0

0 0 0 0 7:1 0 0 78 31:4 0 0 0

0 0 31:4 0 219:6 0

3 2 3 δx 0 6 7 −550 7 7 6 δy 7 7 6 δz 7 0 7  6 7 ð35Þ 7 6 ϕx 7 0 7 6 7 5 4ϕ 5 0 y 10; 000 ϕz

3 2 3 δx 0 6 7 −31:4 7 7 6 δy 7 6 7 0 7 7  6 δz 7 6 7 0 7 7 6 ϕx 7 0 5 4 ϕy 5 219:6 ϕz

ð36Þ

where F, M, δ, ϕ are the force ([MN]), moment([MN·m]), deformations([m]), and rotation angle([rad]). The subscripts x, y, and z correspond to the directions shown in Fig. 3. The Rayleigh damping of the structure is set up by providing the relative damping of 1% for the modes at an eigenfrequency of 0.1 Hz and 10 Hz respectively. 5.2. Hydrodynamic coefficient The hydrodynamic forces on a tubular component per unit length are calculated by Morison's equation as expressed in Eq. (37) [51].

Node number

Brace (element number)

Position description

Fatigue life (years) With gravity

Without gravity

Error

10201 10503 30553 20627 10704 10754 10803 10804

40240 30410 30512 90617 30612 10754 307082 608032

A leg joint at the bottom A leg joint A face joint A conductor support A leg joint A leg joint A leg joint A leg joint

252 124 73 79 43 581 59 29

312 123 74 79 44 540 67 30

24% −0% −1% 0% 2% −7% 14% 3%

This equation is only applicable when the diameter of the structural component d is less than 1/5 of the wave length, which is fulfilled by jacket structures: 1 F ¼ ρ  A  a þ ρ  C m  Ar  ar þ ρ  C D  vr jvr j  d 2

ð37Þ

where ρ A a CM Ar ar CD vr d

is the density of the seawater. is the cross section area of the body. is the component of water particle acceleration normal to the component axis. is the added mass coefficient. is the reference area normal to the structural component axis. is the relative acceleration between water particle and component normal to component axis. is the drag coefficient. is the water particle velocity relative to the component normal to the component axis. is diameter of the component exposed to the sea.

The first item ρ ⋅ A ⋅ a in the right hand side of the equation is wave potential-related Froude–Krylov excitation force, which is the sum of the hydrodynamic pressures acting on the surface of the body. The pressure disturbance due to the presence of the body is taken into account in the second item ρ ⋅ Cm ⋅ Ar ⋅ ar, which is the added mass (ρ ⋅ Cm ⋅ Ar) related force due to the relative acceleration (ar) between the body and the water. In general, this depends on the flow conditions as well as the location of the body. The wave frequency-dependent characteristics of the added mass may be disregarded for the deep submerged bodies, provided the dimensions of the body are smaller than the wave length. Note that both the Froude–Krylov force and added mass force are due to the inertia of the structures and the surrounding waters. The viscous effects are then considered in the third item (drag force) 12 ρ  C D  vr jvr j  d of Eq. (25). This item also indicates a nonlinear relationship between the wave current speed and its reaction forces on tubular components. In this study, the hydrodynamic drag and inertia (CM = 1 + Cm) coefficients are applied according to latest version of NORSOK N003 [52] as illustrated in Table 1. Note that compared to the original version of NORSOK N-003 released in 1999, the inertia coefficient for the latest version of NORSOK N-003 is significantly decreased, while the drag coefficient is slightly increased. This indicates that if the inertia coefficient is a dominant factor in the structural response, fatigue life according to the latest version of NORSOK Standard may then be increased significantly compared to that of the original one.

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

11

Fig. 10. The global behaviour of the jacket structure in time domains at sea state block 7 (Hs = 2.9 m, and Tp = 7.8 s); one year fatigue damage without considering the probability of occurrence 3.380 · 10−2 for block 7.

5.3. Wave modelling A design wave is described by the wave height, modal wave period and direction. In this study, the jacket is analysed for 22 sea states and 8 wave directions for each sea state.

The wave scatter diagram corresponding to the target sea area in the North Sea is divided into 22 blocks and the probability for each sea state is shown in Table 2. The directional probability of the wave is illustrated in Fig. 5. The total number of load cases is then 22 × 8 = 176. From Fig. 5 it is

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J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

Fig. 11. The global behaviour of the jacket structure in time domains at sea state block 19 (Hs = 8.8 m, and Tp = 13.2 s).

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

13

Table 7 The statistical characteristics of the global response of the jacket for sea state blocks 7 (Hs = 2.9 m, Tp = 7.8 s), 11(Hs = 4.6 m, Tp = 8.3 s) and 19 (Hs = 8.8 m, Tp = 13.2 s) respectively. Wave heading: platform south. Sea state block

Block 7

Block 11

Block 19

Items

m k2 k3 k4 m k2 k3 k4 m k2 k3 k4

With gravity load

Without gravity load

Global displacements (m)

Base shear (N)

Overturning moment (N·m)

Global displacements (m)

Base shear (N)

Overturning moment (N·m)

4.8898E − 02 8.0200E − 06 1.3539E − 01 3.1525E + 00 4.9325E − 02 2.3794E − 05 4.0044E − 01 2.8484E + 00 5.1140E − 02 1.4900E − 04 3.1511E + 00 2.3005E + 01

2.0437E + 05 2.4500E + 10 1.1271E + 00 4.2934E + 00 3.2756E + 05 7.0191E + 10 1.2371E + 00 4.4927E + 00 9.3914E + 05 1.1900E + 12 3.3849E + 00 2.0864E + 01

2.8375E + 07 4.8000E + 14 1.1464E + 00 4.3931E + 00 4.5116E + 07 1.3458E + 15 1.2616E + 00 4.6631E + 00 1.2000E + 08 2.1400E + 16 3.7632E + 00 2.4838E + 01

4.7000E − 03 0.0000E + 00 1.1029E + 00 4.0444E + 00 5.8660E − 03 1.8049E − 05 1.4290E + 00 5.8077E + 00 1.3919E − 02 2.0000E − 04 2.8108E + 00 1.5408E + 01

2.0512E + 05 2.4600E + 10 1.1207E + 00 4.2704E + 00 3.2832E + 05 7.0180E + 10 1.2438E + 00 4.5384E + 00 9.4536E + 05 1.2000E + 12 3.3670E + 00 2.0624E + 01

2.8475E + 07 4.8200E + 14 1.1427E + 00 4.3738E + 00 4.5194E + 07 1.3454E + 15 1.2702E + 00 4.7221E + 00 1.2000E + 08 2.1500E + 16 3.7532E + 00 2.4650E + 01

observed that the most dominant wave directions with respect to the probability of occurrence are south, South West and north, while the directions of east, North East and South East show the least probability of occurrence. The JONSWAP spectrum is used to generate the wave energy input. The directional spreading (short wave crest) and the current are not taken into the account in the current modelling as permitted by the NORSOK Standard [52]. 5.4. Marine growth Marine growth is a common designation for a surface coat on marine structures, caused by plants, animals and bacteria. It may cause increased hydrodynamic actions, increased weight and added mass, which may influence hydrodynamic instability due to vortex shedding and possible corrosion effects [52]. Table 3 shows the modelling of marine growth at different elevations on the jacket structure.

5.8. Analysis procedure The fatigue analysis is normally performed using the following procedures: (1) calculate the stress variations during the lifetime; (2) count the stress amplitudes using certain counting methods, in the current paper, rainflow counting method is adopted; (3) calculate the fatigue damage for each stress amplitude range according to the material data (S–N curves) [53]; and (4) sum them up to estimate the total fatigue damage during the structure's entire fatigue life. The total fatigue damage is based on nonlinear dynamic analyses of 22 sea states for one-hour durations from 8 wave directions. In total, 176 simulations are carried out to calculate total fatigue damage. The FE programme USFOS was used to carry out the dynamic analysis in the time domain. In each time step, the hydrodynamic forces are recalculated according to the updated structural deformation results. The computational code FATAL is adopted to count the stress amplitude cycle and calculate fatigue damage. The analysis procedure is summarised in Fig. 6.

5.5. Splash zone The splash zone is defined to be between EL −5.0 m and +7.9 m. 5.6. Buoyancy effect The buoyancy effect is calculated up to sea level. 5.7. Set up of numerical parameters The values for γ in the JONSWAP wave spectrum, the numbers of input(N) at discrete frequencies, and the time step length (Δt) in the dynamic simulation using the HHT-α time integration algorithm are set as 2, 120, and 0.25 s respectively.

6. Modal analysis Modal analysis is focused on the eigenfrequencies and mode shapes. From a physical perspective, an initial excitation of an undamped system will cause it to vibrate and the system response is a combination of eigenmodes, where each eigenmode oscillates at its associated eigenfrequency [55]. When the mass of the structure is large, the stiffness is low, and the operating loads are close to or pass through one or more of the natural frequencies of the structures, or there are transients with a duration close to the half of natural periods of the structure, modal analysis is essential to identifying the condition of the dynamic structural response magnifications due to excitation loads [56].

Table 8 The statistical characteristics of the global response of the jacket for sea state blocks 51 (Hs = 2.9 m, Tp = 7.8 s), 55(Hs = 4.6 m, Tp = 8.3 s) and 63 (Hs = 8.8 m, Tp = 13.2 s) respectively. Wave heading: platform east. Sea state block

Block 51

Block 55

Block 63

Items

m k2 k3 k4 m k2 k3 k4 m k2 k3 k4

With gravity load

Without gravity load

Global displacements (m)

Base shear (N)

Overturning moment (N·m)

Global displacements (m)

Base shear (N)

Overturning moment (N·m)

4.895E − 02 9.574E − 06 − 5.265E − 03 2.7465E + 00 4.956E − 02 2.346E − 05 1.004E − 01 4.145E + 00 4.919E − 02 1.874E − 04 1.954E + 00 1.1895E + 01

1.875E + 05 2.100E + 10 1.128E + 00 4.514E + 00 3.059E + 05 6.835E + 10 1.431E + 00 5.326E + 00 8.917E + 05 8.230E + 11 2.580E + 00 1.2758E + 01

2.599E + 07 4.066E + 14 1.147E + 00 4.584E + 00 4.223E + 07 1.328E + 15 1.443E + 00 5.480E + 00 1.113E + 08 1.433E + 16 2.695E + 00 1.344E + 01

3.842E − 03 7.805E − 06 1.450E + 00 5.893E + 00 5.308E − 03 1.325E − 05 9.222E − 01 3.6339E + 00 1.345E − 02 1.908E − 04 2.644E + 00 1.1674E + 01

1.874E + 05 2.119E + 10 1.145E + 00 4.551E + 00 3.358E + 05 7.205E + 10 1.256E + 00 4.598E + 00 7.597E + 05 6.713E + 11 1.854E + 00 6.426E + 00

2.595E + 07 4.097E + 14 1.168E + 00 4.630E + 00 4.642E + 07 1.389E + 15 1.237E + 00 4.590E + 00 9.322E + 07 1.103E + 16 1.928E + 00 6.702E + 00

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Table 9 The statistical characteristics of the local response for joint 10803 at sea state blocks 7 (Hs = 2.9 m, Tp = 7.8 s) and 19 (Hs = 8.8 m, Tp = 13.2 s). Wave heading: platform south. Sea state block

Block 7

Block 19

Items

m k2 k3 k4 m k2 k3 k4

With gravity load

Without gravity load

Axial force (N)

In-plane bending (N·m)

Out-of-plane bending (N·m)

Axial force (N)

In-plane bending (N·m)

Out-of-plane bending (N·m)

−3.08E + 06 8.35E + 08 3.90E − 01 7.41E − 01 −3.11E + 06 9.18E + 09 −5.59E − 01 8.47E + 00

−3.55E + 05 5.89E + 07 1.35E + 00 5.89E + 00 −3.53E + 05 3.33E + 08 −1.77E − 01 4.60E + 00

2.03E + 04 8.75E + 08 −2.08E − 01 9.49E − 01 −1.16E + 03 3.59E + 09 −2.24E + 00 4.49E + 01

−8.35E + 04 1.22E + 09 2.46E − 01 2.95E + 00 −1.05E + 05 7.58E + 09 −7.76E − 01 11.44E + 00

−1.89E + 04 6.92E + 07 1.28E + 00 5.37E + 00 −1.55E + 04 3.11E + 08 −2.08E − 01 4.72E + 00

−3.18E + 04 8.17E + 08 −2.70E − 01 9.63E − 01 −4.81E + 04 3.37E + 09 −2.33E + 00 4.59E + 01

The eigenvalue calculation is performed according to the system mass and stiffness matrix. Geometric stiffness changes continuously due to changed component forces, and the instantaneous position will also give different transformation matrices between local elements and the global system. By accounting for cases either with gravity or without gravity effects, Table 4 shows the comparison of the first ten eigenperiods of the jackettopside-flare tower structure involving the influence from the hydrodynamic added masses at calm sea conditions. The description of each mode shape and some representative mode shapes are shown in Table 4 and Fig. 7 respectively. Stress “softening” effects described in section 0 can be clearly shown in almost all eigenmodes. Particularly in the lower order of the eigenmodes, the eigenperiods for cases with gravity effects are higher than their counterpart without gravity effects. The higher the order of the eigenmode, the smaller the eigenperiod difference between the two cases. One exceptional case is for the 9th eigenmode, where the eigenperiod with gravity effect is even lower than that without gravity effect. Another exceptional case is where a local vertical vibration at the bottom horizontal frames occurs at an eigenperiod of 1.014 s for cases with gravity effect, rather than that without gravity effects (1.152 s). This is because in structures with large degrees of freedom, even though gravity can induce compression in some structural components, it may also introduce tension in some other components. This is especially applicable to the higher order of eigenmodes, which show more local vibration mode than global vibration mode. Slightly tuning local component stiffness can cause the eigenfrequencies of local vibration mode to increase (due to tension), decrease (compression) or even cross each other. To further investigate the influence due to P-Delta and stress “stiffening/softening” effects separately, the most important three eigenperiods are calculated at different deflection levels of the structure. Fig. 8 shows the time history of the base shear, overturning moment and the resultant horizontal displacement at the centre of gravity (CoG) of the topside under the highest sea state (for the current fatigue analysis) from platform south and west with Hs = 12.8 m, and Tp = 13 s. It is observed that the base shear, overturning moment and horizontal displacement vary almost in phase with each other. Four time instants are selected for further examination: at time t = 2.0 s and 4.0 s, when the global responses are small, and at t = 13.75 s and 19.25 s, when

the horizontal displacement approaches its two peaks. Table 5 lists the calculated eigenperiods at those four time instants. The calculated eigenperiod at different times is almost identical, indicating that, for the current structure being studied, compared to the stress “stiffening/ softening” effects, P-Delta effects are insignificant on tuning the structure's stiffness and eigenperiods. This is mainly due to the fact that the sea state used for this investigation does not induce any plastic structural deformation, i.e. no stiffness degradation due to plasticity, and the horizontal deflection of the structure also remains limited with a maximum value of 0.14 m, i.e. the stiffness change due to geometric nonlinearities is small. However, this conclusion is based on foundation modelling with a constant linear spring system in all six degrees of freedom. P-Delta effects may have a certain influence on tuning the foundation stiffness through the action of overturning moment and vertical loads. Therefore, if the topside deflection is higher under adverse sea state, the structure is more slender, or the nonlinear foundation stiffness is accounted for, the P-Delta effects may then be more relevant. It should be noted that the first two eigenperiods are close to the low values of modal wave period. In addition, the structure also shows large vibrations near the sea surface. Significant dynamic magnification responses are then expected when the structure encounters low wave periods. Examining Table 4, note that the periods for the high order of eigenmodes are more closely spaced than those of the low order of eigenmodes. This may indicate redundancy of the structure.

7. Analysis of the fatigue calculation results The calculated fatigue life at the selected joints (Fig. 9) on the jackets is shown in Table 6. This shows that ignorance of the gravity loads introduces fatigue calculation error by −6% to 24%. The negative sign in the error indicates a decrease in fatigue life and vice versa. This means that for the selected joints, ignorance of the gravity loads may lead to an overestimation of fatigue life by up to 24%. The gravity loads significantly increase the calculated fatigue damage for the joint 10201 close to the seabed and joint 10803 just above the still water surface, while decreasing the calculated fatigue damage at joint 10754 just below the still water surface.

Table 10 The statistical characteristics of the local response for joint 10803 at sea state blocks 51 (Hs = 2.9 m, Tp = 7.8 s) and 63 (Hs = 8.8 m, Tp = 13.2 s). Wave heading: platform east. Sea state block

Block 51

Block 63

Items

m k2 k3 k4 m k2 k3 k4

With gravity load

Without gravity load

Axial force (N)

In-plane bending (N·m)

Out-of-plane bending (N·m)

Axial force (N)

In-plane bending (N·m)

Out-of-plane bending (N·m)

−3.07E + 06 1.47E + 09 5.26E − 03 3–6.20E − 01 −6.73E + 04 1.11E + 09 3.51E − 02 3–5.20E − 01

−3.54E + 05 5.36E + 07 2.28E + 00 8.39E + 00 −1.78E + 04 5.83E + 07 2.12E + 00 7.39E + 00

2.42E + 04 2.89E + 07 5.61E − 01 3 + 4.88E − 01 −2.82E + 04 2.95E + 07 5.61E − 01 3.477E + 00

−3.06E + 06 2.87E + 10 7.32E − 01 6.76E + 00 −4.90E + 04 3.25E + 10 1.16E + 00 6.77E + 00

−3.48E + 05 2.76E + 08 5.66E − 01 3–8.01E − 01 −1.08E + 04 2.32E + 08 5.38E − 01 3–9.20E − 01

2.71E + 04 2.74E + 08 9.84E − 03 4.63E + 00 −2.63E + 04 2.61E + 08 −1.14E + 00 6.55E + 00

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

15

Fig. 12. Time series of the local response (axial force (Dof 1), in-plane bending moments (Dof 5) and out-of-plane bending moments (Dof 6)) for joint 10803 at sea state block 7(Hs = 2.9 m, Tp = 7.8 s). One year fatigue damage without considering the probability of occurrence 3.380 · 10−2 and 3.804 · 10−2 with and without gravity load effects respectively.

8. Statistical analysis The global behaviour (global displacement at the top of the jacket, base shear force, overturning moments with respect to the mud line and the centre of gravity) of the jacket structure in time domains for two sea state blocks 7 and 19 are illustrated in Figs. 10 and 11 respectively. The gravity loads induce deflection of the structure, which encounters the waves through the fluid–structure interaction varying over time. They then influence the variation of the mean stress as well. Naturally, the gravity loads significantly increase the mean value of the global displacement at the top of the jacket, while they influence the displacement amplitude less significantly. For the fatigue of welded joints not sensitive to mean stress, this indicates that the gravity loads may generally have limited influence on fatigue damage in most of the joints on the jacket. Note that the calculations of the base shear force and overturning moment only include wave forces without any structural inertia forces, and are therefore very slightly influenced by the gravity loads. Note that the peak value of the base shear for high sea state block 19 is almost ten times that of low sea state block 7. Similar to the influence of the global displacement, the gravity loads significantly influence the position change of the centre of gravity, while only

slightly influencing the variation amplitude of the centre of gravity in time domains. In order to study the structure's local response statistically and to check the reliability of the calculation results, the global behaviours are subject to further statistical analysis. Tables 7 and 8 illustrate the mean value (m), variance (k2), skewness (k3) and kurtosis (k4) of the global structural behaviour for sea state blocks 7, 11, 19, 51, 55, and 63. The skewness (k3) and kurtosis (k4) are defined as Eqs. (38) and (39): 1 k3 ¼ lim T→∞ T

 # Z T " xðt Þ−ux 3 dt σx 0

ð38Þ

1 T→∞ T

 # Z T " xðt Þ−ux 4 dt σx 0

ð39Þ

k4 ¼ lim

where μx and σx are the mean and standard deviation of a random process x(t), and T is the time interval. Note that skewness characterises the degree of asymmetry of a distribution around its mean. Compared to a Gaussian (normal)

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J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

Fig. 13. Time series of the local response (axial force (Dof 1), in-plane bending moments (Dof 5) and out-of-plane bending moments (Dof 6)) for joint 10803 at sea state block 19 (Hs = 8.8 m, Tp = 13.2 s). One year fatigue damage without considering the probability of occurrence 5.767 · 10−1 and 5.320 · 10−1 (block 19) with and without gravity load effects respectively.

distribution with skewness equal to 0, positive skewness indicates a distribution with an asymmetric tail extending toward more positive values. Negative skewness indicates a distribution with an asymmetric tail extending toward more negative values. Kurtosis characterises the relative peakness or flatness of a distribution compared with the normal distribution. A kurtosis of more than 3 indicates a relatively peaked distribution. A kurtosis of less than 3.0 indicates a relatively flat distribution. The non-Gaussian distribution of the response can be identified by a non-zero skewness and a kurtosis not equal to 3.0 of the response. A larger deviation of the kurtosis from the value of 3 indicates more

significant deviation of the response from that of a Gaussian distribution. Compared to a Gaussian distribution of the stress response, a non-Gaussian distribution can significantly change the overall distribution of rainflow cycles, greatly affecting the probability of occurrence of the largest cycles' amplitude. Particularly for a kurtosis greater than 3.0, the non-Gaussian spectrum attributes great probability of occurrence to the largest amplitudes (with thicker tails in the probability density function diagram) than its Gaussian counterpart, resulting in a shorter fatigue life. The opposite can be expected when the kurtosis is less than 3.0. This difference is particularly evident for large stress amplitude.

Table 11 Statistical characteristics of the local response for joint 10804 at sea state blocks 7 (Hs = 2.9 m, Tp = 7.8 s) and 19 (Hs = 8.8 m, Tp = 13.2 s). Sea state block

Block 7

Block 19

Items

m k2 k3 k4 m k2 k3 k4

With gravity load

Without gravity load

Axial force (N)

In-plane bending (N·m)

Out-of-plane bending (N·m)

Axial force (N)

In-plane bending (N·m)

Out-of-plane bending (N·m)

−3.08E + 06 8.35E + 08 0.390329 3.44072 −3.11E + 06 9.18E + 09 −0.5586 8.470104

−3.55E + 05 5.89E + 07 1.351461 5.888711 −3.53E + 05 3.33E + 08 −0.17709 4.602058

2.03E + 04 8.75E + 08 −0.20849 3.649037 −1.16E + 03 3.59E + 09 −2.23803 17.88835

−8.35E + 04 1.22E + 09 0.245774 2.94603 −1.05E + 05 7.58E + 09 −0.77583 11.438591

−1.89E + 04 6.92E + 07 1.276788 5.364872 −1.55E + 04 3.11E + 08 −0.20779 4.719731

−3.18E + 04 8.16E + 08 −0.27041 3.663945 −4.81E + 04 3.37E + 09 −2.33022 19.89876

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

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Fig. 14. Frequency responses of the axial force (Dof 1) [N], in-plane bending moments (Dof 5) [Nm] and out-of-plane bending moments (Dof 6) [Nm] for joint 10803 for sea state block 7 (Hs = 2.9 m, Tp = 7.8 s).

When gravity loads are considered, the mean value of the global displacement is of course significantly higher than cases that do not consider gravity loads. Conversely, the influence of the gravity on the mean value of the base shear and overturning moments is not that significant. It is also discovered that, under the same wave heading of platform south, at low sea state blocks 7 and 11, the gravity loads tend to slightly increase the variance of global displacement. However, at high sea state block 19, the gravity loads tend to slightly decrease the variance of the global response of the jacket. Unlike sea state 19, the gravity loads at low sea state blocks 7 and 11 tend to significantly decrease the skewness and kurtosis of the global displacement. This means that gravity causes the global displacement to become more non-Gaussian at high sea states and more Gaussian at low sea states. Note that in the current analysis, different types of nonlinearities coexist, and the phenomena observed above are likely to be caused by the “cancelling” effects between P-Delta and other nonlinear effects at low sea states, or by the “enhancing effects” at high sea states. It is also observed the global responses generally become more non-Gaussian as wave height increases. This is due to the combinations of the nonlinearities induced from the variation of water surface causing intermittency of wave loading, the variation of buoyancy forces on components in the splash zone, nonlinear drag forces in Morison's equation, and large structural deformations. Note that joint 10803 has a 14% higher fatigue life if the gravity load effects are not accounted for. The statistical characteristics of its response are also studied as described in Tables 9 and 10 and Figs. 12

and 13. The mean values of the responses with gravity load effects are of course significantly higher than those of cases without gravity load. At lower sea state block 7, the variance of the axial force, in plane bending moments involving gravity load effects are even lower than their counterpart without considering the gravity load's effects. While at high sea state block 19, the variance of the responses is increased due to gravity effects. At low sea state block 7, the gravity load tends to increase the skewness of the axial forces and in-plane bending moments, while decreasing the skewness of out-of-plane bending moments. However, at high sea state block 19, gravity intends to decrease the skewness of all the local responses. The gravity load at sea state block 7 decreases the axial and out-of-plane bending moments' kurtosis, while increasing the kurtosis of the in-plane bending moments. At high sea state block 19, it decreases the kurtosis of all three force components. The gravity loads have more influence on the kurtosis of the axial forces than that of the bending moments. Similar trends can be observed for joint 10804, as shown in Table 11. The local response of joint 10803 is further investigated in frequency domain as shown in Figs. 14 and 15. Generally, two peaks can be identified: one corresponds to the wave modal period; the other corresponds to the structure's natural vibration period. It is noted that the gravity loads have almost no influence on the response at the wave modal period, this tallies well with the observation conclusions of the base shear and overturning moments in section 0. However, their effects can be clearly observed at the structure's natural period. At low sea state block 7, the gravity loads decrease the axial forces and in-

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J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

Fig. 15. Frequency responses of the axial force (Dof 1) [N], in-plane bending moments (Dof 5) [Nm] and out-of-plane bending moments (Dof 6) [Nm] for joint 10803 for sea state block 19 (Hs = 8.8 m, Tp = 13.2 s).

plane bending moment for joint 10803, while increasing the out-ofplane bending moments. While at high sea state block 19, the gravity loads significantly increase the axial forces and in-plane bending moment for joint 10803 and slightly increase the out-of-plane bending moments. Figs. 16 and 17 show the frequency response of the axial force (Dof 1) of in-plane bending moments (Dof 5) and out-of-plane bending moments (Dof 6) for joint 10804 in two sea state blocks — 7 and 19. Similar to that of joint 10803, two major peaks can be identified: one corresponds to wave excitation frequency; the other corresponds to the structure's natural frequency. Gravity does not influence the frequency response at wave excitation frequency. However, it does influence the magnitude of the frequency response corresponding to the structure's natural frequency by slightly increasing the out-of-plane bending moments' response, while significantly decreasing the axial force and in-plane bending moment response. The mean value (m), variance (k2), skewness (k3) and kurtosis (k4) of time domain responses for joint 10804 for the two sea states blocks 7 (Hs = 2.9 m, Tp = 7.8 s) and 19 (Hs = 8.8 m, Tp = 13.2 s) are checked as shown in Table 11. It is clear that the presence of gravity loads will increase the absolute value of the mean stress. However, the relationship trend between the variance of the response and the gravity load effects

cannot be identified. Examining the table, a significant non-Gaussian response distribution can be observed. This non-Gaussian distribution is more significant when the wave height increases from 2.9 m to 8.8 m. This is due to the increase of nonlinearities induced from the waves, wave-structure interactions (wave intermittency and nonlinear drag), and geometrical nonlinearity of the structure. The trend between the skewness or kurtosis and the sea states cannot be identified. At the low sea state, the effects of gravity at joint 10804 seem to enhance the peakness of the response distribution. Meanwhile, at the high sea state and, unlike the response at joint 10803, gravity loads lower the peakness of the distribution. Therefore, the trend from the gravity load on the nonGaussian distribution of the local response cannot be identified. Fig. 17 shows the frequency response of joint 10804 at high sea state block 19. Similar to the response at the low state block 7, the frequency responses still exhibit two peaks corresponding to the wave excitation frequency 0.0758 Hz (13.2 s) and the structure's natural frequency. Again, gravity does not influence the magnitude of the frequency response corresponding to wave excitation frequency, while at the structure's fundamental eigenfrequency, it significantly increases the frequency response of the axial force and in-plane bending moments and slightly increases the frequency response of the out-of-plane bending moments.

J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

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Fig. 16. Frequency responses of the axial force (Dof 1) [N], in-plane bending moments (Dof 5) [Nm] and out-of-plane bending moments (Dof 6) [Nm] for joint 10804 for sea state blocks 7 (Hs = 2.9 m, Tp = 7.8 s).

9. Closing remarks The effects of gravity on the fatigue calculation of offshore structures are mainly attributed to P-Delta and stress “softening” effects. For fixed offshore structures with large degrees of freedom, gravity load generally increases the eigenperiod of vibration mode. However, not only can it induce compression on structural components, but it also induces a tendency to cause tension at some other components locally. At higher orders of eigenmodes with significant local vibrations, slightly tuning the local component stiffness can cause the eigenfrequencies to increase (due to tension), decrease (compression) or even cross each other. Without accounting for the nonlinearities of the foundation, P-Delta effects are insignificant on tuning the structure's stiffness and eigenperiod compared to the stress “stiffening/softening” effects. However, if the topside deflection is higher under adverse sea state, the structure is more slender, or the nonlinear foundation stiffness is accounted for, the P-Delta effects may then be more relevant. The effects of gravity on the fatigue damage assessment are studied on the basis of the nonlinear dynamic response calculation of a typical offshore structure. It is discovered that ignorance of the gravity loads can underestimate the fatigue damage by up to 24%. For a more slender structure, these effects can be even more significant. Gravity significantly

enhances the non-Gaussian trend of the structure's global displacement at high sea states, while slightly decreases the non-Gaussian trend at low sea states. However, its influence on the skewness and kurtosis of the local structural responses (element forces, moments, stresses) may be different from that of the global response. Furthermore, it is also discovered that the variation trend of skewness and kurtosis due to gravity may coexist differently. The sea state with higher wave height increases the mean value, variance, skewness, and kurtosis of the structure of both the global and local structural response. Gravity can have significant effects on the frequency content of the local structural response. With the residual stress on the welded steel joints, for a conservative calculation of fatigue life, the effect of mean stresses are not taken into account and the stress amplitude is always assumed to be tensioned. However, residual stresses may actually relieve themselves by gradually dissipating under sustained cyclic loads. It is recommended that the benefits of this effect with respect to fatigue estimation are considered for a future study. This study is based on an assumption that the foundation of the structure can be represented as a linear spring system in all degrees of freedom, which is also default practice for fatigue analysis by the majority of the design codes and in industry practice. However, in

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J. Jia / Journal of Constructional Steel Research 118 (2016) 1–21

Fig. 17. Frequency responses of the axial force (Dof 1) [N], in-plane bending moments (Dof 5) [Nm] and out-of-plane bending moments (Dof 6) [Nm] for joint 10804 for sea state blocks 19 (Hs = 8.8 m, Tp = 13.2 s).

reality, the additional overturning moment and vertical load induced from the gravity effects on the foundation also change the stiffness of the foundation. Especially under adverse sea states, even though these sea states cherish a low probability of occurrence, the soil plasticity may develop. Therefore, further investigation of gravity effects with a nonlinear model such as a Winkler foundation model for piles is recommended, since this may have a notable effects with respect to both P-Delta and the stress “stiffening/softening” effects. Moreover, after a significant cyclic loading, gaps (cavities) may appear between the pile and surrounding soils, the change in load bearing capacity and load–deflection relationship may also be worthy of investigation. For education purposes, knowledge of stability and vibration are essential for structural and mechanical engineers, and they are strongly related to each other within both theory and abundant practical engineering applications. However, in university education they are almost taught as two separate topics. In addition, plasticity and stability are also correlated (elastic–plastic buckling analysis, for example,) while presented as two individual subjects even in concurrent textbooks. Therefore, it is recommended that students are presented with some basic knowledge about the interactions of vibration and plasticity with the stability subject.

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