Proceedings of OMAE2002: 21st International Conference on Offshore Mechanics and Arctic Engineering June 23-28, 2002, Oslo, Norway
STRUCTURAL RESPONSE OF PIPELINE FREE SPANS
BASED ON BEAM THEORY
Olav Fyrileiv Det Norske Veritas N-1322 Høvik, Norway
Kim Mørk Det Norske Veritas N-1322 Høvik, Norway
One of the main risk factors for subsea pipelines exposed on the seabed is fatigue failure of free spans due to ocean current or wave loading. This paper describes how the structural response of a free span, as input to the fatigue analyses, can be assessed in a simple and still accurate way by using improved beam theory formulations.
In connection with the release of the DNV Recommended Practice, DNV-RP-F105 “Free Spanning Pipelines”, the simplified structural response quantities have been improved compared to previous codes. The boundary condition coefficients for the beam theory formulations have been updated based an effective span length concept. This concept is partly based on theoretical studies and partly on a large number of FE analyses. The updated expressions are general and fit all types of soil and pipe dimensions for lower lateral and vertical vibration modes.
The present paper focus on estimation of simplified response quantities such as lower natural frequencies and associated mode shapes. Hydrodynamical aspects of Vortex Induced Vibrations (VIV) are outside the scope of this paper.
KEYWORDS: Pipeline free span, structural response, beam
theory, natural frequencies, modal stress, fatigue, over-stress
Free span assessment is one of the most important tasks both during design of a pipeline and later during survey, operation and potential intervention. Pipelines encounter free spans when the pipeline is laid on uneven seabed or due to seabed scouring effects. The pipeline may fail due to over-stress caused by its self-weight in combination with environmental forces or by fatigue due to vortex-induced vibrations (VIV) and
direct wave forces. For this reason the free spans need to be assessed, and if necessary corrected, to ensure the integrity of the pipeline.
The costs, which relate to seabed correction prior to pipeline installation and span intervention in later phases, are often significant in terms of total costs of a pipeline project. On the other hand the potential costs (recovery costs and economical loss) related to fatigue failure of a pipeline are enormous. Despite of these aspects, free spans are normally designed applying unduly conservative concepts and often very simple analytical tools.
The DNV guideline no 14 (GL 14) for free spanning pipelines was issued in 1998 and has been used on a large number of projects and gained general acceptance in the industry. This guideline has been reviewed and updated lately to account for technical development and research within this field, to account for experience in applying the guideline and to improve the user friendliness. The new document is issued as a Recommended Practice (RP).
Pipeline free span design involves several disciplines and must consider the following aspects:
· Environmental conditions · Loading mechanisms · Structural response · Acceptance criteria
Given the environmental conditions for a free span, the hydrodynamical loading in terms of VIV is usually addressed by a so-called response model. Here, the reduced velocity which again is given by the flow velocity, the pipe diameter and the natural frequency of the span, is the basic parameter. By determining the natural frequency, the reduced velocity is found and the response model gives the vibration amplitude. Then the stress range giving fatigue accumulation is found from the
so-Proceedings of OMAE’02
International Conference on Offshore Mechanics
and Artic Engineering
June 23-28, 2002,Oslo, Norway
called modal stress, i.e. the bending stress given by the mode shape associated with the natural frequency, and the fatigue accumulation is calculated.
The structural response parameters of a free span, such as natural frequencies and associated modal stress, are often based on simplified beam theory expressions. The response is significantly influenced by factors such as the soil stiffness, the axial force and the static deflection of the span. For this reason, ordinary beam theory expressions will often fail to estimate the structural response with a required accuracy for free span assessment. Thus the end result may be over-conservative in some cases while it becomes non-conservative in other cases.
One alternative to beam theory expressions is clearly use of non-linear finite element (FE) methods. Provided that the modelling of span support conditions and other details are done properly, this method provides reliable and accurate results. The method also provides large flexibility in modelling different aspects like span geometry, span support etc, see Fyrileiv et al. (1998) and Kristiansen et al.(1998). However, FE analysis is time consuming and not very efficient if a large number of spans are to be screened.
Hobbs (1986) addressed the importance of the natural frequency when assessing free spans. His paper was based on a non-linear, numerical solution of the governing differential equations including the effect of soil stiffness at the span supports. However, the effect of the axial force was not accounted for. Park and Kim (1997) also based their work on an analytical solution of the differential equation and gave simplified design criteria for over-stress and on-set of VIV based on dimensionless curves, still without the effect of the axial force. A similar approach was followed by Kapuria et al. (1998) where the effect of the axial force was accounted for. Common for all these methods is that they are based on non-linear, numerical solutions of complex equations. It may be argued that use of FE methods is as easy as this method is well known and is offering more flexibility in defining/modelling the problem.
However, a simple but still reasonable accurate method for estimating all the essential structural response parameters like natural frequencies, associated modal stress and static bending moment is needed for rational screening of free spans in design and during operation of pipelines. In connection with the release of the DNV Recommended Practice, RP-F105, the simplified structural response quantities given in Guideline no 14 (1998) have been improved. The beam theory formulations have been updated based on an effective span length concept. This concept is party based on theoretical considerations and partly a large number of FE analyses. The updated expressions are general and fit all types of soil and pipe dimensions.
This paper describes how the structural response of a free span can be assessed in a simple and still accurate way by using improved beam theory formulations. Comparisons with related studies by Hetenyi (1946) and Hobbs (1986) are also given.
RELATED THEORETICAL STUDIES Introduction
The basic parameters determining the structural response of a free spanning pipeline are illustrated in Fig. 1.
a) Pipeline free span
b) Physical model
Fig. 1 – Free spanning pipeline
Several papers and textbooks discuss the structural response of free spanning beams or pipelines. As an example, Clough and Penzien (1975) give the basic theory for the dynamics of a beam under influence of axial force.
The differential equation for a freely vibrating beam may be written as: 0 t v m x v S x v EI 2 2 e 2 2 eff 4 4 = ¶ ¶ + ¶ ¶ + ¶ ¶ (1)
where v is the lateral deflection, x the axial co-ordinate, t is time, Seff the effective axial force, EI the bending stiffness and
me the effective mass of the beam per length (including mass of
content and hydrodynamical added mass).
The classical solution of this equation involves separation of variables assuming the solution having a form as:
) t ( Y ) x ( ) t , x ( v =j × (2)
where j(x) describes the mode shape and Y(t) describes the time variation.
Now, two ordinary differential equations are obtained: 0 Y Y&&+w2 = (3) 0 EI m EI Seffj ¢¢- ew2j= + j ¢¢¢¢ (4)
where w is the undamped, angular frequency of the beam. The solution in terms of fundamental natural frequency for a vibrating beam with axial force becomes:
÷÷ø ö ççè æ × + × = E eff 2 4 e 1 0 1 C SP L m EI C f (5)
where L is the beam (span) length, PE the Euler buckling force
(=p2EI/L2) and C
in this paper it is shown how these varies for different span lengths and soil conditions for real free spanning pipelines. It may be mentioned that C1=3.56 and C2=0.25 for a fixed-fixed
support condition. Note that the Seff/PE term becomes negative
when the effective axial force is in compression since PE is
defined as positive.
Study by Hobbs
In his paper, Hobbs (1986) puts emphasis on the importance of the natural frequency when addressing free span problems and in particular how the frequency and the boundary condition coefficients are influenced by the seabed elasticity/stiffness at the span shoulders.
Accounting for the elastic soil support at span shoulders and neglecting the effect of the effective axial force, eq. (4)
converts to: 0 EI ) m K ( 2 e = j w -+ j ¢¢¢¢ (6) where K is the soil stiffness.
The problem is now to solve this equation and satisfy the boundary conditions, especially at the span shoulders modelled by elastic springs. After some lengthy manipulations, Hobbs establishes a complex, non-linear equation to be solved to find the constants entering the solution of eq. (4). This non-linear equation may be solved by iterative, numerical methods. However, a graphical solution is represented in the paper and given in fig. 2.
The abscissa in fig. 2 is the logarithm of some sort of relative soil stiffness to pipe bending stiffness:
EI L K× 4
While the ordinate, lL, indirectly gives the natural frequency of the span as:
4 2 e EI mw = l (8)
Fig. 2 – Graphical solution for a free spanning pipeline, Hobbs (1986)
For a given value of log10b, fig. 2 gives the corresponding
lL value. Now as l is known, eq. (8) may be “inverted” to get the angular frequency and, thus, the natural frequency, f0, see eq.
Theory by Hetenyi
The textbook by Hetenyi (1946) treats the theory and solutions for several cases of beams supported on elastic foundation. One case is buckling of a free spanning beam supported on linear elastic foundation (soil). By solving the static differential equation:
0 Kv x v S x v EI 2 2 eff 4 4 = + ¶ ¶ + ¶ ¶ (9)
and applying the proper boundary conditions, the following non-linear equation gives the condition for buckling:
÷ ø ö ç è æ p + p = z 2 cos 1 z y 2 (10) Given b = = EI KL y 4 (11) and E cr P P z= (12)
Here, the value of y is given by the known input parameters and eq. (10) is solved by an iterative approach. z is then given and Pcr, the critical axial force at which the span buckles, is
EFFECTIVE SPAN LENGTH
The main objective of the study described in this paper is to establish simple, but still accurate estimates for the structural response of a pipeline free span. In order to simplify this complex problem, an effective length, Leff, is introduced. This
effective length concept means that the structural response of the pipeline span resting on elastic supports may be calculated as the response of a equivalent pipe of length Leff with
Hobbs (1986) also discusses a similar effective span length. However his effective span length is based on a pinned-pinned support as shown in fig. 3. In the study described by this paper, an effective length based on fixed-fixed supports was found to be superior as a fixed-fixed support is a realistic, asymptotic boundary case for a free span as the span length and the soil stiffness increase.
Fig. 3 – Effective span length, Hobbs (1986).
Considering an effective span length giving the same natural frequency as the real span supported on elastic foundation, eq. (5) may be expressed as (neglecting the axial force term): e 2 eff e 21 0 m EI L 56 . 3 m EI L C f = = (13)
By manipulating eq. (8), the natural frequency may also be expressed as: e 2 e 21 0 m EI 2 m EI L C 2 f p l = = p w = (14)
As both eqs. (13) and (14) give the frequency, they may be used to establish the following expression for the effective span length: L 73 . 4 L 2 56 . 3 L Leff l = l p × = (15)
where lL is found from fig. 2 or by solving the non-linear equation of Hobbs (1986).
A different expression is found when considering the effective span length as the length giving the same buckling load as the real span supported on elastic foundation. Then eq.
(12) may be expressed as:
2 eff 2 2 2 2 eff 2 E cr L L 4 L EI L EI 4 P P z = p p = = (16) or z 4 L Leff = (17)
Now, z is to be found by solving eq. (10).
The effective spans lengths given by eqs. (15) and (17) have been plotted in fig. 4 for a real pipeline case using different soil stiffnesses and span length resulting in different b values. Here a number of finite element analysis results have also been included where the natural frequencies have been calculated and the effective span length calculated by eq. (13). A very good correlation between the FE results and the expression by Hobbs
(1986) is found. Also the effective span length based on the expressions by Hetenyi (1946) shows a good correlation to the other results even though this effective span length relates to buckling not the fundamental frequency.
1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 log10b Leff /L FE results Hobbs Netenyi RP-F105 estimate
Fig. 4 – Effective span length as a function of log10b.
Based on these calculated effective span length values, a new expression for the effective span length is found by curve fitting. The effective span length used in the DNV-RP-F105 (2002) reads: 7 . 2 7 . 2 for for 0 . 1 61 . 0 036 . 0 73 . 4 63 . 0 02 . 1 066 . 0 73 . 4 L L 2 2 eff < b ³ b ï ï î ï ï í ì + b + b + b + b -= (18) where: ÷ ÷ ø ö ç ç è æ + × = b EI ) CSF 1 ( L K log10 4 (19)
CSF is the stiffening effect of any concrete coating on the pipeline, see Fyrileiv et al. (2002) for further details.
As seen in fig. 2, the lL-curve and, hence, the effective span length shows a change in curvature close to the pinned-pinned condition at b= 2.7. Therefore the curve fit expressed by eq. (18) is given by two parts at each side of this shifting point to provide a better fit.
It should be noted that a similar effective span length concept has been used in previous studies, see e.g. Palmer and Kaye (1991). However, in this study a more accurate approximation is made which significantly improves the accuracy of the modified beam theory.
NEW RP-F105 EXPRESSIONS
The following approximate response quantities based on the effective span length concept are given in the new DNV-RP-F105 (2002).
The expression for the lowest natural frequency reads:
÷ ÷ ø ö ç ç è æ ÷ ø ö ç è æ d + × + × + × » 2 3 E eff 2 4 eff e 1 0 D C P S C 1 L m EI CSF 1 C f (20)
where d is the static/steady-state deflection at span midpoint. Hence, the geometrical stiffening effect due to sagging is accounted for by the C3 term. For the horizontal direction C3 normally equals zero. Note that PE is now based on effective span length, i.e. PE=p2EI/Leff2
The modal (dynamic) stress for one diameter maximum deflection amplitude is derived from the mode shape and is given by:
)2 eff s 4 CF / IL L E ) t D ( D CSF 1 C A = + × - × (21)
where Ds is the steel pipe diameter and D is the outer pipe diameter (including any coating). t is the steel pipe wall thickness and C4 is a boundary condition coefficient.
The static bending moment may be estimated by:
÷÷ø ö ççè æ × + × = E eff 2 2 eff 5 static P S C 1 L q C M (22)
where q represents the loading, i.e. the submerged weight of the pipe in the vertical (cross-flow) direction or the drag loading in the horizontal (in-line) direction. This expression is based on the static moment of a span with an approximate correction due to the effective axial force. Such a correction is discussed by Timoshenko & Gere (1961) and is found to give an error less than 2% for C2Seff/PE < -0.6. Limiting the effective axial force in this way means that other sources such as the effective axial force level and the soil stiffness will be the dominating sources of uncertainty and not this approximation itself.
In case the static deflection is not given by direct measurement (survey) or estimated by more accurate analytical tools, it may be estimated by:
÷÷ø ö ççè æ × + + × × = d E eff 2 4 eff 6 P S C 1 1 ) CSF 1 ( EI L q C (23)
The boundary condition coefficients C1 to C6 are given in
Table 1 for different support conditions. The present formulation for the effective span length implies that the values for single spans on elastic supports (seabed) approach the fixed-fixed values, as the effective span length becomes equal to the
real span length (very long free spans and very high soil stiffness).
Table 1 - Boundary conditions coefficients
Fixed-Fixed 3) Single span on seabed
C1 1.57 3.56 3.56 C2 1.00 0.25 0.25 C3 0.8 1) 0.2 1) 0.4 1) C4 4.93 14.1 Shoulder: 14.1(L/Leff)2 Mid-span: 8.6 C5 1/8 1/12 Shoulder: 6 ) L / L ( 18 1 2 eff -Mid-span: 1/24 C6 5/384 1/384 1/384
1) Note that C3=0 is normally assumed for in-line if the steady
current is not accounted for.
2) For pinned-pinned boundary condition Leff is to be replaced
by L in the above expressions also for PE.
3) For fixed-fixed boundary conditions, Leff/L=1 per
4) C5 shall be calculated using the static soil stiffness in the
Note that different effective span length apply in the horizontal and vertical directions and for the static and dynamic response quantities in case of different soil stiffness.
The approximate response quantities specified in this section may be applied for free span assessment provided: · Conservative assumptions are applied with respect to span
lengths, soil stiffness and effective axial force.;
· The span is a single span on a relatively flat seabed, i.e. the span shoulders are almost horizontal and at the same level; · The symmetrical mode shape dominates the dynamic
sponse (normally relevant for the vertical, cross-flow re-sponse only). Here the following limits apply:
L/Ds < 140
d/D < 2.5
Note that these are not absolute limits; the shift in cross-flow response from the symmetrical to the unsymmetrical mode will depend on the sagging and the
level-ling/inclination of the span shoulders. In cases where a shift in the cross-flow response is considered as likely, the struc-tural response of the span should be assessed by using FE analysis including all important aspects;
· Bar buckling is not influencing on the response, i.e. C2Seff/PE > -0.5;
EXAMPLES AND DISCUSSION
The structural response quantities are shown in Fig. 5 to Fig. 12 for a typical 40” pipeline as an example for discussion. The response quantities given by the expression in RP-F105 are compared with the corresponding FE results. Similar comparison has been done for both a 20” and a 10” pipeline and with a similar good correlation between FE results and RP-F105 values.
Fig. 5 shows how the effective span length varies with different soil types (stiffness by RP-F105 values) and span lengths. The FE results are shown with symbols, while the corresponding, fitted values as given by eq. (18) are shown with lines. As seen, the fit is very good giving values close to the FE results. For short spans and especially on very soft soils the effective span length is under-estimated and thus the frequencies will be non-conservative. For the sake of curiosity it can be mentioned that in the extreme case, having a L/Ds of 20,
even the pinned-pinned case will be non-conservative. From the figure it is seen that the boundary conditions for a span will approach the fixed-fixed one as the span length and soil stiffness increase. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 0 20 40 60 80 100 120 140 L/Ds Lef f /L clay-very soft clay-soft clay-firm sand-loose sand-medium sand-dense
Fig. 5 – Leff/L for different span lengths and soil types,
horizontal direction, 40” pipe. FE results shown by symbols and estimated values by solid lines.
In Fig. 6 and Fig. 7 the predicted horizontal and vertical frequencies are plotted relative to the corresponding FE results for 210 cases covering different span lengths, soil types and levels of effective axial force.
It is seen that the estimated in-line (horizontal) frequencies are within ±5% of the FE values except for the shortest spans where the effective span length give an over-prediction in frequency. This is not regarded as a problem as such spans will not become critical for realistic flow velocities.
The same tendency as for the horizontal frequencies are seen for the vertical ones, i.e. the correlation with the FE results are within ±5% except for the spans with a large mid-span deflection. The spans with a mid-span deflection larger than 2.5 times the outer diameter are shown with open symbols, not filled.
Predicted frequency/FE frequency (in-line)
0.8 0.9 1.0 1.1 1.2 0 20 40 60 80 100 120 140 L/Ds
Fig. 6 – Predicted horizontal frequency over FE results for different span lengths, soil types and effective axial force levels.
Predicted frequency/FE frequency (cross-flow)
0.8 0.9 1.0 1.1 1.2 0 20 40 60 80 100 120 140 L/Ds
Fig. 7 – Predicted vertical frequency over FE results for different span lengths, soil types and effective axial force levels.
Predicted dynamic stress/ FE dynamic stress (cross-flow)
0.4 0.6 0.8 1.0 1.2 1.4 0 20 40 60 80 100 120 140 L/Ds
Fig. 8 – Predicted vertical dynamic (modal) stress over FE results for different span lengths, soil types and effective axial force levels.
The predicted modal stresses in the vertical direction are shown in Fig. 8. As for the vertical frequencies, the modal stress estimates are quite good and within ±10% except for the shortest spans and the long spans with a large mid-span
deflection (>2.5D shown as not-filled symbols). The modal stresses in the horizontal direction show a similar correlation with the FE results except that the fit for the long spans are better since the problem with sagging and shift of modes is not relevant in this direction.
Fig. 9 shows the predicted static bending stress relative to the corresponding FE results for the 210 cases. Again a reasonable good correlation is found with a tendency to under-predict the stresses for the shortest spans. However, the static stress enters only the ultimate limit state check. For the range of application for the RP-F105 expressions, the static stress is not a concern unless for relatively long free spans. Therefore, the RP-F105 expression is found to give a good estimate of the static stress.
Predicted static stress/ FE static stress
0.6 0.7 0.8 0.9 1.0 1.1 1.2 0 20 40 60 80 100 120 140 L/Ds
Fig. 9 – Predicted vertical static bending stress over FE results for different span lengths, soil types and effective axial force levels
Fig. 10 shows the predicted static deflections relative to the corresponding. Again a very good correlation is found for span lengths where the deflection contributes to the frequency. Some scatter occur for the shortest spans due to the inherent inaccuracy for the short spans with minor deflections.
Predicted static deflection / FE static deflection
0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 20 40 60 80 100 120 140 L/Ds
Fig. 10 – Predicted vertical deflection over FE results for different span lengths, soil types and effective axial force levels.
The horizontal and vertical mode shapes are shown for various span lengths in Fig. 11 and Fig. 12. Here the pipeline axis co-ordinate has been normalised with the span length such
that the span supports starts at ±1. As seen from the horizontal modes, the effective length decreases as the span length increases. The same tendency is seen for the vertical modes.
An interesting effect is shown for the vertical mode as the span length approach 120 metres. Then the mode shape curvature close to the span supports becomes smaller, and for the span with a length of 140 metres, the lowest, symmetrical mode shape becomes suppressed and an unsymmetrical mode shape appears. This will cause the associated frequency and modal stress to increase as seen in Fig. 7 and Fig. 8.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 -3 -2 -1 0 1 2 3
Normalised span coordinate
No rm alised m o d e sh ap e L = 20 m L = 40 m L = 60 m L = 140 m
Fig. 11 - Horizontal mode shapes.
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -3 -2 -1 0 1 2 3
Normalised span coordinate
alised mode shape
L = 20 m L = 40 m
L = 60 m
L = 140 m L = 120 m
Fig. 12 - Vertical mode shapes.
A comparison between the natural frequencies found by FE analyses and the new RP-F105 estimates using simplified boundary conditions, is shown in Fig. 13. The FE analyses were performed at a real 40” pipeline case and observed free spans using the measured seabed configuration and span lengths. The following comments to the FE analyses applies:
· No concrete coating stiffening effect was accounted for. · A low added mass coefficient, Ca = 1.1-1.2, was applied.
· An unrealistic low dynamic soil stiffness (300 kN/m/m) was applied.
In order to compare the results, the same added mass and soil stiffness were used in the RP-F105 expressions for the horizontal frequency as in the FE analyses.
As seen from the figure, the boundary condition for a free span may be considered as close to pinned-pinned for short spans. When the span length increases, the effective boundary condition approaches asymptotically the fixed-fixed case. This tendency is confirmed by the expression for the effective span length. The reason for the scatter, although not very significant, as shown in the FE results is partly different gaps and thereby different added masses, and partly the fact that all the real spans are not idealistic single spans on a flat seabed but may be affected by small neighbourhood spans. Anyway it may be concluded from this comparison that the updated expressions in RP-F105 provides, for this case, natural frequencies of the same accuracy as a non-linear finite element analysis.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 30 40 50 60 70 80 90 100 L f0 RP-F105 fixed-fixed pinned-pinned
Fig. 13 - Natural frequencies from FE analyses compared to estimates based on simplified boundary conditions.
To further demonstrate the effects of the updated structural response parameters in RP-F105, a simple case have been analysed using a 40” gas export pipeline with a D/t of 42. The pipeline is coated with 100 mm concrete and is located on 80m water depth in a typical North Sea environment. As the considered stretch is far away from the inlet, the temperature of the gas is equal to the ambient temperature. The operating pressure is 130 bar. The seabed consists of loose sand, which causes free spans due to scouring. A design life of 50 years is assumed in the calculations.
As seen from Fig. 14, the boundary conditions have a significant effect on the calculated fatigue life. It is further seen that for this case, the updated structural response quantities from RP-F105 give fatigue lives almost as a fixed-fixed span somewhat depending on the span length. The maximum allowable span length for a design life of 50 years typically increases from 62 to 71 metres changing from the pinned-fixed boundary condition often used in the industry to the one given in RP-F105. 1 10 100 1000 40 50 60 70 80 90 100 Span Length (m) F atig u e life (year s) RP-F105 Fixed-fixed Pinned-fixed Pinned-pinned
Fig. 14 – Fatigue lives applying different boundary conditions.
SUMMARY AND CONCLUSIONS
A short introduction to the structural response quantities given in the new DNV-RP-F105 for free spanning pipelines has been given. The following observations and comments apply: · A set of simple expression for structural response
quanti-ties, like natural frequencies, modal stresses, static bending moment and deflection, is given.
· The structural response quantities are based on traditional beam theory and the use of an effective span length. · The new expressions have shown to give good
correspon-dence with FE results.
· A comparison between different boundary conditions re-vealed that the new DNV-RP-F105 expressions allow sig-nificantly longer spans than the traditionally used pinned-fixed condition for sandy conditions.
The main conclusion from the case study presented in this paper is that the new DNV-RP-F105 expressions give structural response quantities at the same level of accuracy as FE analyses for the lowest vibrations modes normally associated with fatigue analyses of free spans. However, this requires that the spans are located on a relatively flat seabed and are not interacting with other spans. In addition the effective axial compression force must be relatively low so that buckling effects are not dominating the response.
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