A method is proposed for VI V fatigue assessment of pipeline spans and risers by direct scaling of model test results. S caling approximations serve to extend the range of applicability of a limited number of VIV test data. Also , scaling is applied only for the dynamic portion of th e response, with the static portion calculated by analysis, which then provides the conditions for scaling. Ultimately using the basic and extended scaling approximations described, the problem of a simply supported span, on uniform pipe subject to consta nt current velocity is approximately characterized by a single dimensionless parameter. As a result, for any (“prototype”) span to be assessed, any VIV test with similar end conditions (e.g. simply supported) can be scaled to yield the prototype current v elocity for which similarity is satisfied. To facilitate this, test data from recent, well -instrumented VIV tests on simply supported spans are processed yielding convenient and physically meaningful normalized parameters describing the test conditions an d response. Fatigue damage rates from the scaling method are compared to those from the DNVGL -recommended assessment procedure and those from the frequency domain version of Vivana. Keywords: riser, pipeline, VIV, scaling, similarity, fatigue 1 Introductio n In a 1979 review article on VIV, Sarpkaya [ 1] predicted that “It is a research area which will continue to challenge the imagination of both the experimentalist and the theoretician for generations to come.” Researches ’ imagination does indeed continue to be challenged 42 years later , with no end in sight, especially when multi -mode response is involved: the need for VIV tests to further cover cases of practical importance remains , especially for higher Reynolds numbers [ 2]. Approximate methods , either by modal analysis and an empirical response function (as in [ 3]) or by (complex) frequency domain analysis with empirical excitation and added mass functions [ 4-8] still miss important effects, and sophisticated ones including 3D CFD are still not generally accepted, in part because validation studies (such as [ 9]) do not yet include the full range of conditions for which the simulation tool might be used in practice , and good predictions for one case, do not necessarily mean they are also good for another. As a result, considerable uncertainty remains about VIV amplitude predictions, and even more about the fatigue damage from such vibrations. To better understand this uncertainty in VIV fatigue damage, a new fatigue assessment method based on extended scaling of model test results is developed . This requires model tests that are qualitatively similar to the prototype to be assessed, e.g. in that the shape of the current velocity profile is similar , or any strake coverage and gaps therein are geometrically similar . Th e method relies on scaling approximations that replace certain assumptions in exsisting assessment methods. The scaling approximations include the well -known (Von Kármán) beam theory for moderately large deflections with constant effective axial force, wh ich is also used in analysis methods, the principle of local action, that the hydrodynamic force at any location along the pipeline depends on the motion at that location only, which has also been extensively used, though not in the way applied here, an ap parently new approximation whereby the requirement for similarity in the axial force may be relaxed, as an approximation, where there is a clear dominant wavelength of the response , and others. Whilst it is not possible to verify the validity of the scalin g approximations with existing experimental data, what is possible and done in this paper is to (a) assess the accuracy of the structural scaling approximations by calculating dimensionless indic ators, (b) calculate convenient non dimensional parameters representing three test programs carried out at Marintek (now SINTE Ocean) with high mode response and heavily instrumented pipes whereby the scaling method is easily applied to any 2 simply supported span, with bare, uniform pipe subject to constant current along the length , and (c) compare the scaling predictions to those from existing analysis methods for an example prototype pipe. If VIV assessment is seen as the art of combining experimental results with theory to estimate fatigue damage , with a spectrum of methods ranging from fully experimental to more fully theoretical, then this paper falls more on the experimental side, requiring experiments closer to the real thing, to more directly use the experimental results , rather than using them in calibrate ap proximate methods. Generally, “Buckingham’s Π Theorem” [ 10,11] is cited for dimensional analysis, although he was not the first to use it. The theorem applies for a relationship between “physical quantities” which Buckingham calls an “equation”, but here is called a function , as in practice the function may be evaluated by interpolation between experimental data points . Quoting Buckingham’s terms, t he premise to his theorem is that if a “complete” set of the “physical quantities” is used, then the funct ion does not depend on the units used, even when the arguments to the function are only the numbers without their units, provided that those units are consistent , i.e. all units used are derived from the same basic ones . This implies that not any function is physically admissible . Buckingham’s Π Theorem then provides and justifies the method to reduce the physically admissible function to its core relationship between dimensionless variables, which are fewer than the original physical quantities by the nu mber of independent (or basic) dimensions involved, which for VIV problems are 3 (e.g. length, time, and mass ). If the function is to be determined empirically, s uch reduction is crucial to limit the scope of the required experimental investigation , and i t has been extensively used in designing and interpreting VIV tests (see for example [2,12]). However, t he choice of dimensionless variables is not unique . Here a combination of dimensional analysis, and approximations is used to reduce the number of dimensionless parameter s, select ing ones that are ph ysically meaningful and are stable in the sense that that the dimensionless output parameters (i.e. those representing results of the tests) do not change very much with changes in the dimensionless input parameters (i.e. those controlled by the design and executio n of the test) . In particular, a normalized fatigue -equivalent bending strain or curvature ϵ̃e is defined and calculated , which varies only weakly as a function of the normalized input parameters , and from which the fatigue damage rate for a prototype pip e can readily be determined. Thus, ϵ̃e is a good parameter to report for VIV test programs in which bending strains are measured, as is done here for recent well -instrumented high-mode test s carried out at Marintek (currently SINTEF Ocean). In contrast to [13], which seeks to explain changes in VIV response amplitude in terms of a normalized equivalent damping coefficient c* on the “power -in” zone of the pipe, which accounts for hydrodynamic damping in the “power -out” zone as well as structural damping everywhere, this effort is aimed defining a complete set of parameters to predict the fatigue damage rate. Indeed, the bare pipe tests considered all have low c* values, since there is no “power out” zone with hydrodynamic damping. “Power in” and “power out” zones are the result of features that must be included in the model test to satisfy similarity. Despite limiting test data considered to those with low c* values, considerable variability remains, and it is this variability that is examined in this paper. 2 Nomenclature 2.1 Terms “Arch action” refers to the phenomenon described in [ 14] whereby static displacements combined with axial restraint at the ends increases the natural frequencies, especially for mode 1. The “dynamic component” or “dynamic portion” of a time series is the amount by which it differs from its average va lue in time during the steady -state VIV. The “dynamic amplitude” of a time series is √2 times the standard deviation. For a purely harmonic signal that is also equal to the amplitude of the sinusoidal signal. The “logarithmic standard deviation” is define d as the standard deviation of the natural logarithm of the quantity in question. For small values, it is approximately equal to the coefficient of variation. “Mode 1” for a simply supported span refers to the vibration mode with a single lobe. Similarly , “Mode 2” refers to a mode with 2 lobes, etc.. Due to arch action, Mode 1 may have a higher natural frequency than Mode 2. 3 “Normalized” parameters are dimensionless parameters that arise from the scaling and are the same for model and prototype. The ben ding strain, for instance, is a dimensionless parameter, but not a normalized parameter in the scaling used. “Pipe” for the VIV tests means a flexible composite cylinder. The “prototype” refers to the pipe, span or riser to be assessed for VIV fatigue, as opposed to the model tests performed to measure the VIV response. “Reference vibrations” (used to assess the accuracy of scaling approximations in Appendix C) are defined to be vibrations of amplitude D, at the Strouhal wavelength (2L st) and fr equency (f st). “Scaling assumptions” or “scaling approximations” refer to Assumptions ( 1) to (9), listed in Section 2. “Selected tests” refers to those selected in Section 6.1 based on indicators of validity of the accuracy of the scaling approximations. The “static component” or “static value” refers to the average value in time when it is applied to a measured quantity. When applied to derived parame ters, it refers to a value calculated using the static values as input. “Straked” means strake -covered. The “theory for moderately large deflections” is the same as the Euler -Bernoulli beam theory, except that a nonlinear term is included in an approximat e expression for the axial strain at the centroid of the beam. See Eq. (39) for the strain -displacement relation, and Eq. (5) for the equilibrium equation derived from it via the principle of virtual displacements. It is the beam theory equivalent of the Von Kármán plate theory. 2.2 Symbols (.) any variable written as (.) = ( .̃)(.) 1. (.̃) normalized value of (.), i.e. a tilde denotes the normalized value. (.)1 unit value of (.), i.e. a subscript 1 denotes the unit value. afat constant that determines the fatigue resistance according to N cf = a fat/Sn where N cf is the number of cycles to failure, S the stress range, and n another fatigue resistance constant. cas still water added mass coefficient, added mass per unit mass of water displaced by the pipe; cas = 1 based on potential theory without seabed proximity effe cts. cd drag coefficient; for VIV tests, this is determined based on static solution from Appendix B, using the measured axial forces under gravity loads only, and the time -averaged axial force under flowing conditions, assuming that c d is constant along the length of the pipe. ceff effective mass coefficient; c eff = m eff/mdw = c pipe + c as. cp velocity of sound in the water. cpipe mass ratio, c pipe = m pipe/mdw. Ck flexural -stiffness -proportional damping coefficient (contribution to bending moment from damping is C k times that rate of change of curvature). C̃k normalized flexural -stiffness -proportional damping coefficient. D outer pipe diameter including coatings. E Young’s modulus for material in which fatigue damage is calculated. EI pipe flexural rigidity including contribution of coatings, if any. EIeff effective flexural rigidity at zero effective axial force to match the stiffness at the dominant wavelength λ; see Eq. (21). fh hydrodynamic force per unit length acting on the pipeline. 4 fnst,arch , fnst natural frequency of the symmetric mode for which the natural frequency is closest to the Strouhal frequency , with and without including arch action, respectively. fst Strouhal frequency f st = S tU/D = 1/T st. g acceleration due to gravity (9.81m/s2), except that in Appendix B, it denotes the increase in length of the pipe over the span due to transverse displacements with ends axially fixed. Iv’ϕ’, Iϕϕ used in Appendix B only to denote integrals over the span of what appears as the subscript; see Eqs. (34) & (35). kx stiffness of axial spring at one end of the VIV test models considered. The other end is assumed to be fully fixed axially. L span length, or any reference length that is similarly defined for model and prototype. L̃ normalized span length, L̃ = L/x 1 = L/L st. Lea effective length for axial stiffness. As in [ 14], this is chosen such that when the pipe becomes longer by an amount Δg due to transverse displacements at the span considered, the corresponding change in axial force is ΔN = EA Δg/L ea. See E q. (32) for the case when the pipe is axially restrained at one end, and restrained by an axial spring of stiffness k x at the other. Lst Strouhal length, used as unit length (x 1) in the normalization adopted, and given in Eq. (10). mdw mass per unit length of fluid displaced by the pipe, m dw = πρD2/4. meff still-water effective mas s per unit length of the pipe, m eff = m pipe + c as mdw. mpipe mass per unit length of the pipe, content and any coatings mpw mass per unit length of the pipe wall including any coatings, but not the content of the pipe (for the VIV test pipes, this is equ al to m pipe) n either the exponent in fatigue resistance curve according to which the increment in fatigue damage η per cycle at strain range Δ ϵ is Δη = (Δ ϵ/ϵfr)n or the mode number of a simply supported span, defined as the number of lobes of the mode considered. In the former case, the API X’ curve is used here for which n=3.74. nst mode number for the symmetric mode for which the natural frequency is closest t o the Strouhal frequency f st. N static effective axial force during VIV; constant with respect to x and t; includes effect of all quasi -static loadings including the static component of the drag force due to current velocity U; for VIV tests N is calculat ed as the average over time (see Assumption ( 6)). Ndyn dynamic amplitude of axial force as measured during VIV test. Ñ normalized effective axial force; see Eq. (12). Ng effective axial force under gravity loads alone (at zero current velocity). Nseq stiffness -equivalent axial force, axial force for which a cable (with zero flex ural stiffness) has the same resistance to sinusoidal displacements at the Strouhal half wavelength as the pipe with flexural stiffness EI and axial force N. qd static drag force per unit length on the pipe, q d = ½ ρ U2 D c d . qg submerged weight per un it length of the pipe, q g = (c pipe – 1) m dw g. q vector of hydrodynamic force per unit length acting on the pipe (in y -z plane) excluding the inertial term associated with the still -water added mass. q̃ normalized hydrodynamic force per unit length excl uding the inertial term associated with the still -water added mass, see Eq. (4). Rϵ radius from the center of the pipe cross section to the location where the bending strain is determined. 5 S stress range considered for fatigue loading. St dimensionless parameter that defines the selected normalization; the normalizations are valid for any value of this parameter; taken to be equal to the Strouhal number. t time. t1 unit value for time, taken to be the Strouhal period, i.e. t 1 = T st = D/(S t U). t̃ = τ normalized tim e, τ = t̃ = t/T st . Tst Strouhal period, used as unit value for time, T st = t1 = D/(S tU). Texposure exposure time during which fatigue damage accumulates; model test time during which approximately steady -state VIV response conditions prevail. T̃exposu re normalized exposure time T̃exposure = T exposure /Tst = number of cycles at the Strouhal frequency. U current speed; in the y direction, i.e. horizontal and normal to the pipeline. Ũ normalized current speed, Ũ = UT st/D . u axial pipe displacement component, i.e. component in the x direction. udyn dynamic part of axial displacement component u. v static displacement of the pipe in the static plane under the action of gravity force q g and drag force q d. v static part of transverse displacements, including a vertical (z) component due to gravity and a lateral (y) one due to the drag force. w transverse displacement vector in y -z plane, w=w(x,t); it can be either the total transverse displacement, or the dynamic component; see the paragraph following Eq. (5). w transverse displacement component (displacement in the y or z direction); used in equations that are valid for both y and z dir ections. w’,w’’, etc. first, second, etc. derivatives of w=w(x,t) with respect to x. ẇ̇̇̇ second derivative of w=w(x,t) with respect to t. w̃ normalized transverse displacement vector in y -z plane, w̃=w/D, w̃=w̃(x̃,t̃). w̃ y or z component of w̃ (in an equation that is valid for both the y and z components). w̃’,w̃’’, etc. first, second, etc. derivatives of w̃=w̃ (x̃,t̃) with respect to x̃. w̃̇̇̇̇ second derivative of w̃=w̃(x̃,t̃) with respect to t̃. x axial coordinate along the pipeline, chosen such that the span covers the interval x ∈ (0,L). x1 unit value of the axial coordinate, taken to be the Strouhal length, i.e. x 1 = L st. x̃ normalized axial coordinate x̃=x/x 1=x/L st. y lateral (horizontal) co ordinate, transverse to the pipeline. z vertical coordinate, increasing upwards. ΔN a change in effective axial force. Δϵ strain range; maximum less minimum strain over a full cycle. Δη increment in fatigue damage η due to a cycle at strain range Δ ϵ. ϵ bending strain in the in line or cross flow direction. ϵ̃ normalized bending strain or curvature, ϵ̃ = ϵ/ϵ1 = w̃’’ (when normalized, curvature and bending strain are equal). ϵ1 unit value for bending strain, ϵ1 = R ϵD/L st2. 6 ϵe equivalent strain ampl itude for which the fatigue damage rate for cycles at amplitude ϵe at the Strouhal frequency (f st=1/T st) matches the actual fatigue damage rate; see Eq. (18). The repor ted values are for the location where the dynamic strain amplitude is the highest. ϵ̃e normalized fatigue -equivalent bending strain or curvature; see Eq. (19). ϵfr constant in S -N fatigue resistance curve Eq. (15); represents strain range at which Eq. (15) predicts failure in one cycle; related to a fat by a fat = (Eϵfr)n. ϵt total dynamic strain including axial as well as bending contributions. η fatigue damage due to bending in the in line or cross flow direction; failure occurs when η reaches 1. η̇ mean fatigue damage rate, for test data this is obtained by Rainflow counting, as described in Appendix A, i.e. it is the average value over time; the values reported in this paper are those at the location where the dynamic strain amplitude is the highest; this serves as an approximation to the location of the maximum fatigue damage. η̃̇ normalized fatigue damage rate from Eq. (17). η̇1 unit value of fatigue damage rate from Eq. (17). λ dominant half wave length for the pipeline response, see Section 5. ν kinematic viscosity of fluid surrounding the pipe. θ angle from vertical of the plane in whi ch pipe hangs under the action of the gravity force q g and the (static) drag force q d (see Appendix B). ρ density of fluid surrounding the pipe. ψ relative error indicators for the scaling assumptions/approximations developed in Appendix C; different ψ values are distinguished by subscripts; ψ ≪ 1 when the scaling approximation is good. ψarch,1 relative increase in natural frequency for mode 1 due to arch action, defined as the natural frequency including arch action divided by that without arch action minus one. (Here the “mode 1” refers to the single lobe mode, even when, due to arch action, its natural frequency is higher than for mode 2.) ψarch,st relative increase in natural frequency due to arch action for the symmetric mode for which the natura l frequency not including arch action is closest to the Strouhal frequency fst = St U/D. ζ modal damping ratio for a sinusoidal mode with half wavelength L st based on the flexural - stiffness -proportional viscous damping included in the formulation. 3 Scali ng Approximations and Formulation Figure 3.1 Span Coordinate System For simplicity, the scaling is first developed for bare pipe of constant cross section over the length of the span shown in Figure 3.1 subject to a constant current velocity along its length . The purpose of scaling is to determine the VIV response and fatigue damage rate for a prototype pipe from VIV tests on a model pipe. For this, the static portion of the response for the prototype pipe is determined by 7 analysis, using a drag coefficient that accounts for the effect of VIV. The scaling is then used for the dynamic portion of the respo nse only. The basic scaling assumptions are those used to derive the basic scaling result of this Section . They leave 3 normalized parameters to define a structurally -undamped, simply supported span. Extended scaling assumptions are developed in Section 5, but already stated in this Section , serve to further reduc e the number of normalized problem parameters from 3 to 1. Basic Hydrodynamic A ssumptions/ Approximations : 1) The water (or other fluid) is incompressible and fully characterized by its density ρ and kinematic viscosity ν. There is no cavitation or air entrainment. 2) The behavior is not sensitive to the viscosity of the fluid. I.e. the Reynolds numb er effect may be neglected , when the problem is nondimensionalized in such a way that the viscosity only appears in the Reynolds number. 3) The assumption of local action : The hydrodynamic force at any location is a function of the lateral and vertical displ acement history w at that location only and is not affected by the motion w at other locations along the span . Basic Structural Assumptions/Approximations: 4) The pipe remains linearly elastic, and the Euler -Bernoulli’s beam approximation that plane sections remaining plane and normal to the axis of the pipe applies. Flexural -stiffness -proportional viscous damping is included. 5) Rotations of the pipeline remain small compared to 1 radian. This applies to static as well as dynamic rotations. 6) The axial force is constant along the length of the pipe, and d ynamic changes in the axial force are neglected . I.e. arch action [14] is neglected. (Such arch action is mostly associated with the first vertical mode , but to a lesser extent it can also occur for higher modes. It requires axial restraint at the shoulders of the span and sag under gravity loads .) Also changes in axial force due to geometrically nonlinear effects such as those considered in [15] are neglected. (The y are also not included in DNV’s recommended span assessment practice [ 3].) 7) The axial inertia within the span length is neglected. The accuracy of the approximations resul ting from the above basic scaling assumptions is discussed in Appendix C. Although data suggest that Reynolds number do have an effect on VIV response [ 2,16], it is generally not practical to carry out model tests with high mode response at the right Rey nolds number. Thus, Assumption ( 2) is more a matter of necessity, than justified by data or physical principles. In practice some allowance for Re ynolds number e ffects not present in the scaled test may be appropriate. See data fitting efforts in Section 6.2 as well as Appendix C for further discussion of Reynolds number e ffects. Extended Assumptions/A pproximations : In addition to the above basic scaling assumptions, the extended scaling assumptions that follow are used in Section 5 , but not required for the basic scaling results of this section. 8) Certain effects associated with the pipe mass are neglected . This amounts to the same approximation as neglecting the effect of the mass ratio in the response functions of [ 3], and allows a model test for one mass ratio to be applied for another. 9) If a single half wavelength λ dominates the response , the stiffening effect of axial tension can be replaced by equivalent flexural rigidity and vice versa. This allows a model test at a di fferent value of the normalized axial force Ñ to be applied. In this section , the basic scaling results are derived using Assumptions (1) to (7) only. First consider a forced vibration test in which a rigid cylinder undergoes a uniform motion w = D w̃(t/t1) in uniform flow of velocity U in the y direction (i.e. horizontal and normal to the cylinder). Here , t1 should be regarded as any time scaling parameter . Changing it makes the normalized motion w̃(.) run faster or slower. By the assumption of local action , the corresponding hydrodynamic force per unit length fh does not depend of the length of the cylinder tested, but it does depend on the following: the dimensionless 8 function w̃(.) that describes lateral as well as vertical motion, and the parameters ρ, ν, D, t, t1 and U. Applying dimensional analysis (Bucking ham’s Π Theorem), this means that the normalized force f̃h = fh/[mdw D/t12] depends on ly on the dimensionless function w̃(.), the normalized time t̃ = t/t1 and the normalized current velocity Ũ = Ut1/D. In the above, mdw = πρD2/4 is the mass per unit length of fluid displaced by the pipe. The hydrodynamic force also depends on Reynolds number UD/ν but that effect is neglected by Assumption (3). Thus , f̃h = f̃h(t̃), where the function f̃h(.) depends on w̃(.) and Ũ. The normalized motion w̃(.) can be any motion, such as the figure -of-8 or C -shaped motions observed in the VIV tests. There will be different motions w̃ (.) at different locations x along the pipeline, but this scaling result can be applied to any of them. To apply Assumption ( 8), the hydrodynamic force fh is further decomposed into a linear term -mdw cas ẇ̇̇̇ that arises from the irrotational flow solution, plus a nonlinear term q. Thus, fh = q – mdw cas ẇ̇̇̇ (1) where c as is the added mass coefficient from the irrotational flow solution, which is c as = 1 in absence of seabed proximity effects, and ẇ̇̇̇ denotes the second derivative of w with respect to time t. In normalized form, this equation can be written as f̃h = q̃ – cas w̃̇̇̇̇ (2) where q̃ = q/[mdw D/t 12] is the normalized nonlinear component of the hydrodynamic force, and w̃̇̇̇̇ is the second derivative of w̃(.) with respect to its argument t̃. It follows that, for a given value of c as, q̃ = q̃(t̃), where the function q̃(.) depends only on w̃(.) and Ũ. Applying this and the approximation of local action (Assumption (2)), for a motion w(x,t)=D w̃(x/x 1,t/t1) (3) the corresponding hydrodynamic force can be written as fh(x,t) = m dw (q̃(x/x1,t/T) D/t 12 – cas ẇ̇̇̇) (4) where the function q̃=q̃ (.,.) depends only on the function w̃(.,.) and the normalized current velocity Ũ=Ut1/D. Further simplification is achieved by the structural Approximations (4) to (7). With these, the equation of motion for the lateral and vertica l displacements w reduces to EI w’’’’ – N w’’ + C k ẇ’’’’ + m pipe ẇ̇̇̇ = fh ∀ x ∈ (0,L) (5) where EI is the flexural stiffness, N the effective axial tension , and C k a damping coefficient associated with flexural deformations of the pipe. Eq. (5) applies for the dynamic portion of the response w=w(x,t) and the corresponding hydrodynamic force per unit length fh which the fluid exerts on the pipe. A similar equation applies to the static portion, except that the inertia and damping term is absent, and the static load also includes the gravity force acting on the pipe in addition to the static p ortion of the force fh exert ed on the pipe by the surrounding fluid . It follows from Assumptions (4) to (7) and momentum balance in the axial direction that N is constant with respect to location x along the span as well as in time t. N is determined based on static analysis considering gravity forces and hydrodynamic drag. In the above, “the static portion” refers to the time -averaged value and the dynamic portion is what remains. The increase in drag force due to VIV should also be included in the static analysis. 9 The effective axial force N includes a contribution from the hydrostatic external pressure. This accounts for changes in area exposed to external pressure under bending deformations that should not be included in fh as well. Substituting Eqs. (3) and (4) into Eq. (5) to write the governing equation in terms of the normalized functions w̃(x̃, t̃) and q̃(x̃,t̃), one obtains (EI D/x 14) w̃’’’’ – (N D/x 12) w̃’’ + (C k D/(t 1 x14)) w̃̇̇’’’’ + (m eff D/t 12) w̃̇̇̇̇ = (m dw D/t 12) q̃ (6) which applies for all x̃ ∈ (0,L̃). W hen applied to w̃, the primes and dots denote differentiation with respect to the normalized axial coordinate x̃ = x/x 1 and time t̃ = t/t 1, respectively. See Nomenclature for explicit definitions. Equation (6) is valid for any t1 and x 1. Different choices of these unit values lead to different normalizations. To facilitate physical interpretation of the normalized parameters that will arise, t1 and x1 are taken to be the Strouhal period and length, T st and L st, respectively, defined by t1 = T st = D/(S tU) (7) (π/L st)4 EI + (π/L st)2 N = m eff (2π/T st)2 (8) Thus L st represents the length of a simply supported span for which the still -water natural frequency is equal to the Strouhal frequency S t U/D. The 2nd equation above is a quadratic in (π/L st)2, which always has one positive root given by (π/L st)2 = {[N2 + 16π2 EI m eff/Tst2]1/2 – N}/(2EI) (9) so that x1 = L st = π (2EI)1/2/{[N2 + 16π2 EI m eff/Tst2]1/2 - N}1/2 (10) Simplifying (6) for the normalization defined by Eqs. (7) to (10) gives {4 ceff/[π2(1 + Ñ)]} (w̃’’’’– π2Ñ w̃’’) + ceff w̃̇̇̇̇ = q̃ – (4/π5) C̃k w̃̇̇’’’’ ∀ x̃ ∈ (0,L̃) (11) where Ñ = N/N 1, N 1 = π2EI/L st2, L̃ = L/L st , C̃k = π3 Cκ ceff/[EI T st (1+Ñ)] (12) In the above C̃k is a normalized damping parameter that reflects the flexural -stiffness -proportional damping included in the formulation. This parameter has been defined in such a way that for a simply supported span of length L st, C̃k is the Scruton mass -damping parameter , which is referred to as the “stability parameter” and denoted by “K S” in [ 3]. The modal damping ratio for a mode with half wavelength L st is ζ = π2 cpipe C̃k. It follows from Eq. (7) and the definition of the normalized current speed Ũ following Eq. (4) that Ũ = 1/S t (13) The above results are valid for any value of St. Only the interpretation of T st and L st as the Strouhal period and length relies on S t being taken to be the Strouhal number. Here S t=0.2 is used as a reference value throughout . This results in Ũ=5. Although and empirical relation between Strouhal and Reynolds number is well -documented [17], the same value of S t must be used for model and prototype for the scaling laws developed here to be valid. If Reynolds number effects are important, the only way to properly account for them is to perform the model test at the Reynolds number of the prototype. 10 Now suppose a model test is performed for given values of the input parameters D, EI, Ck, L, m eff, N, U and ρ. Then, the normalized solution w̃= w̃(x̃,t̃) derived from this model also applies to a prototype with different values of the input parameters provided that the normalized input parameters Ñ, ceff, L̃ and C̃k are the same for model and prototype. Of course similarity in the boundary conditions must also be satisfied. For the cases covered in Table 3.1, this means the normalized stiffness coefficients k̃ or k̃ij in Table 3.1 must be the same for the model and the prototype. For real spans with the pipe supported by soil at the shoulders, the bou ndary conditions are complex: they are nonlinear and inelastic due to soil plasticity and possible loss of contact . However in VIV tests, the ends are often simply supported . That is the case for all VIV tests reported in this paper. For the tests with strake -covered pipes and a gap in the strake coverage at midspan, the response tends to decay towards the shoulders , so the exact boundary conditions may not be that important for such cases. For the undamped, pinned -pinned or fixed -fixed case , there are 7 problem input parameters, leading to only 3 normalized problem input parameters. The means that an experimental program to cover cases of engineering interest needs to cover a 3 -dimensional region of interest in the normalized parameter space. It requir es far fewer experiments to cover the 3-dimensional normalized parameter space than to cover the 7 -dimensional space of actual input parameters. Here “cover” means fill the space with a grid of experimental points, that are sufficiently closely -spaced tha t one can interpolate between the points. For example, if 10 points are sufficient to capture the range of values of engineering interest of each parameter, it requires 103 tests to cover the normalized parameter space instead of 107 to cover the actual p arameter space. However, it still takes a larger number of tests than are currently available to cover the 3 -dimentional normalized space. A further reduction to just one input parameter , the normalized span length L̃=L/L st, is achieved in Section 5 using Assumptions (8) and (9). A similar scaling result could be derived purely by dimensional analysis using Eq. (5) only to identify the “complete” list of “physical quantities” [10] for which a “physical equation” exists as ρ, ν, D, m pipe, EI, Ck, N, L, U i.e. 9 “physical quantities” which can be reduced to 6 normalized parameters L̃, Ñ, C̃k, cpipe, DU/ν, D/L. The only difference is D/L, which the more detailed analysis above ha s eliminated using local action. Having obtained the normalized solution , the bendin g strain can be calculated from ϵ = w’’ R ϵ = ϵ1 ϵ̃ , ϵ̃ = w̃’’ , ϵ1 = RϵD/L st2 (14) For the fatigue resistance, Miner’s rule is applied, and it is assumed that the S -N curve is linear when plotted on a log -log scale. This means that fatigue failure occurs when the fatigue damage reaches 1, and the increment Δη in fatigue damage per cycle at strain range Δ ϵ can be written as Δη = (Δ ϵ/ϵfr)n (15) where ϵfr and n are constants that define the S -N fatigue resistance curve. The total fatigue damage η is obtained by summing the damage for each cycle identified by a Rainflow algorithm, and the mean fatigue damage rate can be written as η̇ = Σ rainflow (Δϵ/ϵfr)n /Texposure = η̇1 η̃̇ (16) where Σ rainflow denotes the sum over all strain ranges Δ ϵ identified by the rainflow counting algorithm used1 over an exposure time T exposure , η̇1 and η̃̇ are the unit value, and normalized value, respectively, of the fatigue damage rate η̇, given by 1 The rainflow algorithm used for the experimental data reported here is “Algorithm 1” from [ 25], with details of its application given in Appendix A. 11 η̇1 = [ϵ1/ϵfr]n /Tst, η̃̇ = Σ rainflow Δϵ̃n /T̃exposure (17) where Δϵ̃ is the range of the normalized strain ϵ̃ = w̃’’, and T̃exposure = T exposure /Tst. Note that ϵ is a dimensionless but not a normalized parameter. For the same normalized solution , there can be different values for the bending strain ϵ for the model and the prototype. Whereas η̃̇ is a properly normalized fatigue damage rate, it has two disadvantages when it comes to presenting the test results in normalized form: (i) the numbers turn out to be large, and (ii) they are sensitive to the fatigue exponent n. Those drawbacks can be overcome by defining a fatigue -equivalent strain amplitude ϵe such that the fatigue damage rate for strain cycles of constant amplitude ϵe at the Strouhal frequency is that same as the actua l fatigue damage rate η̇. Thus, η̇ = (2ϵe/ϵfr)n/Tst , ϵe = ½ ϵfr (Tst η̇ )1/n (18) Or in normalized form, η̃̇ = (2ϵ̃e)n , ϵ̃e = ϵe/ϵ1 = ½ ( η̃̇ )1/n (19) For a give n VIV test, this fatigue -equivalent normalized strain amplitude ϵ̃e also depends on the fatigue exponent n, but the dependence is much weaker than for η̃̇. As such, as an approximation, ϵ̃e, values may be used to calculate fatigue damage rates in prototype pipes for slightly different values of the exponent n for which the ϵ̃e values are derived from the model tests. 12 Description Formulation Normalized Formulation Pinned -Pinned w = 0, w’’=0 at x=0,L w̃ = 0, w̃’’= 0 at x̃=0,L̃ Fixed -Fixed w = 0, w’=0 at x=0,L w̃ = 0, w̃’=0 at x̃=0,L̃ Coupled linear rotational and transverse spring shoulders with same properties for the shoulders on both sides At x=0: –N w’ + EI w’’’ = k 11 w – k12 w’ –EI w’’ = k 12 w – k22 w’ At x=L: N w’ – EI w’’’ = k 11 w + k 12 w’ EI w’’ = –k12 w – k22 w’ where k ij are the components of a 2 × 2 stiffness matrix for the boundary at the span shoulder . The point directions are 1 for displacements w, and 2 rotations –w’. (Here the same shoulder stiffness matrix is assume d for the vertical and lateral directions with the resistances for these directions assumed to be uncoupled. A more general linearized boundary stiffness could also readily be defined, but results in a greater number of distinct normalized stiffness coeff icients.) At x̃=0: –Ñ w̃’ + w̃’’’ = k̃11 w̃ – k̃12 w̃’ – w̃’’ = k̃12 w̃ – k̃22 w̃’ At x̃=L̃: π2 Ñ w̃’ – w̃’’’ = k̃11 w̃ + k̃12 w̃’ w̃’’ = –k̃12 w̃ – k̃22 w̃’ where k̃11 = k 11 Lst3/EI k̃12 = k 12 Lst2/EI k̃22 = k 22 Lst/EI Continuation of the pipe model beyond the shoulder on a bed of springs of stiffness k. Governing equation for soil -supported region beyond the shoulder becomes Eq. (5) with an additional term k w on the left hand side. The additional term of the left hand side of Eq. (11) becomes k̃ w̃, where k̃ = kLst4/EI. Table 3.1 Examples of simple boundary conditions that can be applied at the shoulder of the span and their normalization. For similarity between model and prototype , the normalized boundary parameters k̃ or k̃ij must be the same for model and prototype. 4 Partially Strake -Covered Pipe So far the basic scaling results ha ve been derived for bare pipe of a uniform cross section along the length of the pipe. However , the same applies for partially straked pipe, provided the strake coverage, and the strakes the mselves are similar for model and prototype. For instance if strakes with a pitch of 15D and a height of 0.25D cover the entire span except a gap of 2 5% of the span length centered at midspan, the same must be the case for the model. In this case the para meters D, EI, m pipe and m dw are chosen to reflect the bare pipe, whereas c as is allowed to change as a function of location, so that in any straked portions of the pipe it accounts for any additional added mass due to the strakes. I t can also account for the increase in diameter due to the strakes and the mass of the strakes themselves. All arguments of Section 2 still apply, except that c as and c eff now vary as a fu nction of location along the pipe. 5 Further Simpli fications If one wants to assess VIV for a prototype and has a model test with the same values of L̃, C̃k, ceff, and Ñ, no further approximations are required. The further approximations described in this section allow the VIV test to be used even if only L̃ and C̃k match but not ceff and/or Ñ. This is based on Assumptions ( 8) and ( 9) from Section 2. 13 First c onsider c eff and Assumption (8). The mass ratio must satisfy c pipe ≥ 1 or the pipe would float to the surface. Typical mass ratios for gas lines are in the range cpipe ∈ (1.1, 1.4). Further, in absence of seabed proximity effects, c as = 1. This means that the effective mass coefficient ceff = c pipe + c as is in the range c eff ∈ (2.1, 2.4). The extremes of this range differ by less than 10% from its middle. On this basis alone, a model test in this c eff range provides a reasonable approximation any prototype in the same range. Further, the normalized equation of motion Eq. (11) contains c eff only on as a factor mult iplying all terms on the right hand side. The right hand side is zero for undamped free vibration response. The normalized frequency of that response is not affected by c eff. The terms that include energy input and dissipation are on the right hand side . These are also unaffected by c eff in the sense that ceff will not affect the balance between energy input and dissipation. These observations suggest the effect of c eff on the normalized solution is limited. (Note that the effect of mass on VIV via th e natural frequency is accounted for even when the effect of c eff on the normalized solution is neglected.) Finally, neglecting the effect of c eff on the normalized solution corresponds to neglecting the effect of the mass ratio on the normalized VIV respo nse amplitude. Th is is the approximation adopted in DNV’s Recommended Practice [3] after extensive calibration efforts using experimental data . It suggests that th e experimental data also show a limited effect of c eff, or at least that it is conservative to use the estimated response for mass ratios near 1 for higher mass ratios. ( Pipe-in-pipe systems for high pressure deepwater fields could have m ass ratios well above c pipe = 1.4.) Indeed, tests reported in [ 18] do show a considerable reduction in response for high mass ratios, except at the peak of the resp onse when plotted as a function of reduced velocity. See Fig. 3 of [ 18]. This peak also corresponds to point where the Strouhal frequency coincides with the natural frequency of the span. Assumption ( 9) eliminate s the similarity requirement for the normalized axial force Ñ. That requires an approximation that is valid when the response has a clear dominant wavelength . If the axial tension is so high that the effect of flexural rigidity is comparatively small even for the shorter wavelengths of the response , this will be a good approximation even if several wavelengths contribute significantly to the response. Whereas Assumption (8) (regarding c eff) is also in [ 3] and, as a result, is much used in practice, the authors are not aware of any previous statement or use of Assumption (9). To use Assumption ( 9), it is necessary to estimate the dominant half wavelength (denoted by λ) from the model test data. Different ways are used to estimate this from the available model test data, as described in Section 8.2. Having obtained λ for the model, the prototype value is readily determined since λ/L and λ̃=λ/Lst must be the same for model and prototype. If the half wavelength λ is indeed dominant, both the axial force N and the flexural rigidity EI for the prototype can be adjusted in such a way that (i) the correct resisting force is developed for sinusoidal displacements at the dominant wavelength and (ii) using the adjusted values of N and EI (denoted by Nadj and EI adj) for the prototype , the corresponding normalized axial force Ñ for the model and prototype are the same. For a displacement sin(πx/λ) , the corresponding transverse force per unit length from the first two terms in Eq. (5) is [EI (π/λ)4 + N (π/λ)2] sin(πx/λ). This is the same as one gets at zero effective axial force with a flexural rigidity given by EIeff = EI + N ( λ/π)2 (20) The requirements (i) and (ii) above yield the following adjusted values for the adjusted flexural rigidity EI and effective axial force N: EIadj = EI eff/( λ̃2 + Ñ), N adj = Ñ π2EIadj/Lst2 (21) This adjustment is carried out for the prototype using the value of Ñ for the model. All calculations for the prototype to determine the motion w=w(x,t) are carried out with the adjusted values of EI, and N. This also yields the curvatures w’’ and corresponding bending strains ϵ = R ϵ w’’. The unadjusted EI and N are used to calculate the bending moments EI w’’ and the static solution under gravity and drag loads . 14 6 VIV Test Data 6.1 Test Definition and Selection To experimentally verify the basic scaling approximations requires VIV test data for which the normalized parameters L̃, Ñ and c pipe are the same, while other parameters ( such as D/L) are different . In lieu of a dedicated test program, such data are not available . Instead, available data from well - instrumented , high -mode tests are used. Although this does not lead to an experimental verification of the scaling approximat ions, it does lead to a range of normalized values for the dynamic, and normalized fatigue -equivalent strain amplitudes, ϵ̃e, which are useful in estimating fatigue damage rates for practical cases, and the uncertainty therein. The tests considered are hig h-mode VIV tests on simply -supported spans performed at Marintek (now SINTEF Ocean). The spans are “towed” in still water to simulate the current. Three different test programs are considered: “EM” tests sponsored by ExxonMobil described in [ 19] with data available from [20] on a vertical riser model that undergoes a slightly curved path in the still water, as the rig rotates about a vertical axis, “OL” tests sponsored by the Ormen Lange project lead by Norske Hy dro [21], and on “Pipe 1” to “Pipe 3” , with different diameters, sponsored by Shell [ 22,23]. Key test parameters shown in Table 6.1 together with ranges of results. Although some of the programs included cases with “sheared current” (i.e. towing speed changing along the length of the model by using a rotating rig), as well as various VIV suppression devices , here attention is focused on tests without supp ression devices in which the current is constant along the length of the span. The models are hinged at the ends for both axes of bending, but restrained against torsion and transverse translation. Axially they are fixed at one end, and at the other the tension is maintained by an axial spring. As a result , the static tension increases under the drag loads from N g to N, and this measured change is used according t o the formulation in Appendix B to estimate the angle θ from vertical at which the pipe hangs, and the effective drag coefficient, c D, assuming that the drag is constant along the length of the pipe. The Shell pipes also have 500kg clump weights at the ends of the model, to stabilize the ends. The hinges at the ends are between the cl ump weights and the pipe . They only allow relative rotations in the two bending directions. The rig design principle is that , in lieu of enough stiffness of the supporting structure, the mass of the clumps is high enough so that the natural frequency for vibrations of the clumps is low compared to the VIV frequencies, and thus the clumps suppress transverse movement at t he ends. However , in the axial direction , the pipe is relatively stiff, and this gives rise to a natural frequency involving the clumps on each side moving in opposite directions with alternating extension and contraction of the pipe. For that mode, the natural frequency can fall in the range of the VIV frequencies. How this might affect the behavior is further considered in Appendix C . All tests are heavily instrumented, and include measurement of the bending strain at many locations along the pipe and the axial force at one end of the pipe. In some cases accelerometers are also included from which displacements can be determined at several points along the pipe. However, since displacement measurements are not available for all cases, attention here i s focused on the strains , which are also the most relevant parameter for fatigue assessment. For each test , the part of the record over which a steady state appears to have been achieved is selected, and the results are presented in terms of the static value (which corresponds to the average in time taken over the selected steady -state portion of the record ), and the dynamic amplitude (which is defined as √2 times the standard deviation of the steady state portion of the time series). If the response is purely harmonic this dynamic amplitude is equal to the amplitude of the sinusoidal variation. The average fatigue damage rate η̇ is determined by rainflow counting (as described in Appendix A) over the steady state portion of the record. This is performed at the cross flow and in line location where the dynamic amplitude of the strain is the largest. (The maximum fatigue damage rate and the maximum dynamic amplitu de of the strain do not necessarily occur at the same location, but to reduce processing effort , the adopted approach can be considered to provide an approximation to the spatial maximum of the average in time of the fatigue damage rate .) 15 From the experime ntal results, the normalized parameters of Section 2 can readily be calculated. Further dimensionless parameters ψ defined in Appendix C that give an indication of the relative errors arising from the scaling approximations are calculated. Of these two are used to eliminate data for which such errors are judged too large. Firstly data for which the measured dynamic axial force is judged too high are excluded. For t his, the dynamic axial force ratio given by ψN = N dyn/Nseq (22) is used, where Ndyn is the dynamic amplitude of the axial force as determined from the VIV test records, and Nseq is given by Nseq = N + π2 EI/L st2 (23) and represents the stiffness -equivalent axial force , for which the stiffness to restore out of straightness as the Strouhal wavelength is the same for a cable under tension N seq as for the pipe under tension N . Only data with ψ N ≤ 0.1 are included. Secondly , ψarch,st, quantifies the effect of arch action on the symmetric mode for which the natural frequency not including arch action is closest to the Strouhal frequency. (Antisymmetric modes are not affected by arch action. ) It is defined by ψarch,st = fnst,arch /fnst – 1 (24) where f nst,arch and f nst are defined as the natural frequencies of the symmetric mode for which the natural frequency not including arch action is closest to the Strouhal frequency in still water with and without considering arch action, as calculated based on [ 14] as described in Appendix B , respectively. Only data with ψ arch,st ≤ 0.1 are included. I.e. if arch action increases the natural frequency by more than 10% the test is excluded. Data exclusions based on ψ N and ψ arch,st are illustrated in Figure 6.1. Further, data are excluded for which the calculated2 static rotation at the ends under flowing conditions exceeds 0.4 radians , as illustrated in Figure 6.2 for the data not excluded by the ψ N and ψ arch,st criteria. The intent of the exclusions is that the remaining data should satisfy Assumptions (5) to (7), at least approximately. Of 111 VIV tests on bare pipe, 73 remain after the exclusions according to the criteria in Figure 6.1 and Figure 6.2. Except for Figure 6.1 and Figure 6.2, all other plots of test data and results in Table 6.1 cover only the included tests. The normalized span length and axial force values, L̃ and Ñ, respectively for the included tests are plotted in Figure 6.3. Although one could design a test program in which L̃ and Ñ are varied independently, that is not the case for the available data: For the OL and EM tests , only the current velocity is varied from one test to the next. As the velocity increases, the Strouhal half wavelength L st decreases . This causes L̃ to increas e and Ñ to decrease. The latter is only partly compensated by the increase in axial force due to the increased drag. For the (Shell -sponsored) Pipe 1 to 3 tests , the axial force under gravity alone is not the same from test to test on the same pipe. A s a result , the points in Figure 6.3 for each of these pipes do not fall on a smooth line. The Reynolds numbers covered by the tests are shown in Figure 6.4, which also show how the Reynolds number changes simultaneously with L̃ within each test series, as ch anges are made to the “towing” speed for a given pipe. Figure 6.3 also shows that there no test series on different pipes with the same L̃ and Ñ values, except for tes ts on rough and smooth versions of Pipe 3 at ( L̃,Ñ)=(3.9,109). The closest it gets for different pipes is shown within the box in Figure 6.3, where one Pipe 1 test is close to several Pipe 2 tests. 2 Calculated as described in Appendix B, but where the measured static axial force under flowing conditions does not exceed that under still water conditions, a drag coefficient of c d = 1.4 is used. 16 Figure 6.1 Exclusion of tests for which the dynamic axial force exceeds 10% of the stiffness -equivalent axial force N seq or arch action increases the natural frequency of the symmetric mode for which the natural frequency not including arch action is closest to the Strouhal frequency by more than 10%. A third criterion, not shown on this figure is that the calculated static rotations at the ends under flowing conditions should not exceed 0.4 radians. Figure 6.2 Static rotations for test data not excluded by the criteria of Figure 6.1. 17 Figure 6.3 Calculated normalized span lengths and axial forces, L̃ and Ñ, respectively for the included test data (on bare pipe under uniform current with ψarch,st ≤ 0.1 and ψN ≤ 0.1) Figure 6.4 Reynolds numbers for included test data 18 Sponsor Ormen Lange Exxon Mobil Shell Pipe 1 Pipe 2 Pipe 3 Span Length, L (m) 11.413 9.63 38 38 38 Pipe Hydraulic Diameter, D (mm) 32.6 20 12 30 80 Length/Diameter Ratio, L/D 350 482 3167 1267 475 Mass Ratio, c pipe 1.37 2.22 1.74 1.54 1.14 Maximum current velocity U tested (m/s) 0.28 2.24 3.45 2.51 1.60 Corresponding maximum test Reynolds Number (× 1000) 7.97 39.2 36.3 65.9 112.5 Axial Rigidity, EA (MN) 5.61 2.83 2.57 7.83 7.83 Flexural Rigidity, EI (Nm2) 203 135.4 16.1 572.3 572.3 Stiffness of Axial Spring at one end, k x (kN/m) 20.35 1.593 0.780 21 21 Factor by which axial flexibility is increased due to end spring, L ea/L 25.1 185 87.6 10.8 10.8 Mass of clump weights at the ends of the model m clump (kg) none none 500 500 500 Number of Tests included (those with ψ N ≤ 0.1 and ψ arch,st ≤ 0.1, a total of 99 tests) 4 10 19 27 13 (1 smooth + 12 rough) Range of Normalized Span Lengths, L̃ 2.6-3.2 1.4-7.5 14-33 3.0-18 2.0-9.3 Range of Normalized Effective Axial Force values, Ñ 0.96-1.2 1.2-24 7.9-30 2.6-57 25-387 Range of calculated static rotations at ends (radians) 0.11-0.11 0.01-0.28 0.05-0.39 0.03-0.31 0.03-0.28 Range of Normalized fatigue - equivalent strain amplitudes, ϵ̃e , for cross flow direction above and in line below 4.3-6.1 3.4-4.7 3.5-5.4 2.5-5.5 6.4-13 7.0-13 6.1-11 4.4-9.7 8.7-13 7.4-12 Table 6.1 Model VIV test parameters and ranges of key reslts. The data shown are the same withing the test series corresponding to each column of the table. 6.2 Test Results The results of the static solution of Appendix B are presented in Figure 6.5 and Figure 6.6. For these , the effective drag coefficient c D is determ ined from the measured change in axial force arising from the flow. The drag coefficients thus obtained can only be accurate if this change i n axial force is large compared to the axial force measurement error. Indeed, the calculation of c d is not only a ffected by axial force measurement errors , but also by thermal expansion and any creep that occurs between the times of the two axial force measurements . For this reason , large errors can be expected when N/N g is close to 1. Indeed, the data in Figure 6.5 are only complete for N/N g ≥ 1.32. For 1 < N/N g < 1.32 there are additional points ranging from c D = 0.56 to c D = 18.1 that fall outside the c D range of the plot, and are judged to be unreliable. Further, the calculation of c D is based on the assumption that it is constant along the length of the pipe, whereas in reality i t may vary along the length of the pipe together with the amplitude of VIV . 19 The maximum normalized dynamic strain amplitudes ϵ̃ are shown in Figure 6.7. More directly relevant to fatigue damage are the normalized fatigue -equivalent strains, ϵ̃e, which are calculated for the same locations where the dynamic amplitude of the bending strain is highest. These are shown in Figure 6.8 with L̃ values, and in Figure 6.9 with Ñ values. It can be seen that the fatigue -equivalent strains tend to be bit higher than the dynamic amplitudes. This is further illustrated by looking at the ratio of strains Figure 6.10: The fatigue -equivalent strain is up to a factor of 1.6 higher than that dynamic amplitude of the same strain, but for two of the EM tests, the fatigue -equivalent strain is below its dynamic amplitude. Therefore it is better to use fatigue -equivalent strains from VIV tests than dynamic amplitudes for fati gue assessments. The scatter in Figure 6.8 and Figure 6.9, suggests that neither L̃ nor Ñ on its own is sufficient to predict the normalized fatigue -equivalent bending strain ϵ̃e. If the basic scaling assumptions hold, then ϵ̃e=ϵ̃e(L̃,Ñ,cpipe), i.e. the normalized equivalent bending strain is a function of the normalized span length, the normalized effective axial force, and the mass ratio. Assuming a linear relationship between the logarithms of these quantities yield s the following optimized fits (in the sense that the sum of the squares of the logarithm of the ratio actual/predict ed values is minimized) : ϵ̃e = 4.90 L̃0.152 Ñ0.157 cpipe-0.593 for cross flow ϵ̃e = 2.72 L̃0.215 Ñ0.214 cpipe-0.244 for in line (25) The ratio of actual/fitted value is plotted in Figure 6.11. Compared to Figure 6.8 and Figure 6.9 it is seen that this fitting reduces the scatter : The logarithmic standard deviations of the ratio of actual/fitted ratios are 0.1 6 and 0.17 for the cross flow and in line fits, respectively, compared with a logarithmic standard deviations for ϵ̃e itself of 0.32 and 0.35 , respectively. (The fitting ensures that the geometric average of the actual/fitted ratios is 1.) Although, i t is conceivable that by using a different functional form a better fit might be achieved, the nature of the remaining scatter in Figure 6.11 indicates otherwise. If a smooth function ϵ̃e=ϵ̃e(L̃,Ñ,cpipe) that fitted the data well existed , the test/predicted ratios in Figure 6.11 would also vary smoothly within each test program, much like the Ñ values vary smoothly with L̃ in Figure 6.3. Considering only the data for which ( L̃,Ñ) falls in the box shown in Figure 6.3, i.e., (L̃,Ñ) ∈ [11,18.5] × [5,10], there remains considerable scatter : the logarithmic standard deviations of the ϵ̃e values in that box is 0.15 for cross flow, and 0.16 for in line , i.e. about the same as for the fit of Eq. (25). Further, in that box, 12 of the 13 data points are for Pipe 2, and only one for Pipe 1. I.e. almost all data come from the same pipe and test setup. Efforts to refine the fitting further risk overfitting, espe cially considering that only 5 different pipes are involved, for each pipe essentially only the current velocity is varied, and the fits in Eq. (25) already involve 4 parameters adjusted to optimize the fit . For this reason, we rejected an attempt to also include the Reynolds number in the fitting process. It achieved a slightly closer fit, at the cost of the fitted ϵ̃e decreasing with increasing Re ynolds numbers, which is against the trend documented in [2,16]. Indeed , by considering 2 of the 5 test rigs considered here, the neural network fit in [16] also showed decreasing response with increasing Reynolds number, but after const raining their fit optimization so that the response increases with increasing Reynolds numbers, they suggest that “t he cross -flow vibration amplitude was found to increase with Reynolds number in the same vibration mode but decrease suddenly as the vibrati on mode shifted to a higher number after which point it will continue to increase with Reynolds number until the next mode shift occurs. ” Here no attempt is made to achieve that sort or resolution with such limited data, and indeed the authors believe tha t additional complicating factors are likely at play, as alluded to in the discussion of scaling approximations in Appendix C. There is no lack of possible explanations for the scatter in the data: that the function ϵ̃e=ϵ̃e(L̃,Ñ,cpipe) is not smooth, tha t one or more of the scaling assumptions are violated, or a combination of both. The accuracy of the scaling approximations is discussed in Appendix C. It remains that fatigue damage from VIV is difficult to predict, especially when several modes are exc ited, and testing limitations typically do not allow carrying out the tests at the desired Reynolds numbers. To assess whether a much -used industry span assessment method might provide better predictions of fatigue damage than Eq. (25), each of the VIV tests is also analyzed according to F105 [ 3] as described in Appendix D, witho ut using the safety factors in F105 , but including arch action in the F105 calculation unless otherwise noted . Based on the fatigue damage rates calculated by F105, the 20 corresponding normalized fatigue equivalent strains ϵ̃e are calculated and the ratio o f ϵ̃e values from the test divided by those from F105 are plotted in Figure 6.12. Clearly F105 is not a good predictor of the fatigue -equivalent strain amplitude ϵ̃e. This is also reflected by the ranges shown in Table 6.2. They indicate that for in line, F105 is always conservative, but for cross flow, it may be non -conservative b y up to a factor of 2.33 according to the selected available test data. The logarithmic standard deviations of the test/F105 ratio are 0. 38 and 0. 27, for cross flow and in line, respectively, are more than the 0.16 and 0.17, respectively, for the simple fit of Eq. (25). Even using ϵ̃e = 7.66 for cross flow irrespective of test conditions more accurately predicts the test data than F105 with a logarithmic standard deviation of 0.32 instead of the 0.38 for F105 . Of course, although using F105 for high -mode response (i.e. high L̃ values) is “allowed”, F105 has not been developed for truly high mode response. The multi -mode formulations in F105 are calibrated on the Ormen Lange data some of which are included here. Indeed, for the selected Ormen Lange Data, the logarithmic standard deviation of the test/F105 ratio of fatigue - equivalent strains is only 0.14 for cross flow an d 0.13 for in line, as per the last row of Table 6.2. For higher L̃ values, F105 is seen to be more conservative, perhaps to cover the uncertainty arising from limit ed calibration effort in that domain. Test / F105 Fatigue -Equivalent Strain Ratio Statistic Cross Flow In Line Maximum Value 0.36 0.27 Minimum Value 2.33 0.96 Geometric Mean Value 0.78 0.48 Standard Deviation of the Natural Logarithm (“Logarithmic Standard Deviation”) 0.38 0.27 Geometric Mean Value for Ormen Lange Data only 1.01 0.83 Logarithmic Standard Deviation for Ormen Lange Data only 0.14 0.13 Table 6.2 Statistics for ratio of fatigue -equivalent strain amplitude from test divided by that calculated by F105 without safety factors, based on included cases (as defined in Section 6.1) with arch action included in the F105 calculation Wherever the test/F105 ratios are presented, arch action has been included in the F105 prediction. However, it does not make much difference if arch action is included or not, since limited arch action is one of the criteria for test data selection in Section 6.1. Specifically due to arch action , the geometric mean value of the actual/F105 ratio of ϵ̃e values increases by 0. 79% for cross flow and decreases by 0.02% for in line , and the logarithmic standard deviation decreases by 0. 33% for cross flow and by 0.06% for in line. (Arch action affects the in line as well as the cross flow directions, since un der the steady state force from the current, the pipe hangs in an inclined “static plane”, as explained in Appendix B and D.) Further remarks in regard to F105 are: 1) If the structural model used in the F105 calculations satisfies the basic structural scaling approximations, it will produce the same results when applied to the model or the prototype provided that L̃, Ñ, and c pipe is the same. The scaling result that ϵ̃e only depends on L̃, Ñ, and c pipe also applies to the F105 calculation. 2) Further to the above, the sensitivity of the F105 calculation to c pipe is small, as it only affects certain calculate d oscillation frequencies used for fatigue cycle counting only a nd not the amplitudes of displacement and strain. 21 Figure 6.5 Drag coefficients calculated from the change in axial force from N g under gravity only to N, the static component under flowing conditions. Figure 6.6 Inclination θ of the static plan plane in which the pipe hangs under the action of gravity and a drag force that is assumed to be constant along the length of the pipe . θ = 0° means the pipe hangs vertically with negligible drag whereas θ → 90° means it hangs horizontally under the drag force which must be very large compared to the gravity force. Where arch action is significant, the vibration modes are in the static plane and normal to it, rathe r than in line and cross flow . See Appendix B. 22 Figure 6.7 Maximum values of the normalized dynamic bending strain amplitude Figure 6.8 Normalized fatigue -equivalent bending strain amplitudes calculated for the locations where the dynamic bending strain amplitude is highest , shown as a function of L̃ 23 Figure 6.9 Normalized fatigue -equivalent bending strain amplitudes calculated for the locations where the dynamic bending strain amplitude is highest, shown as a function of Ñ Figure 6.10 Ratio fatigue -equivalent / dynamic amplitudes for the bending strains at the locations where the dynamic amplitude is highest 24 Figure 6.11 Ratio of actual/fitted values of the (normalized) fatigue -equivalent bending strain based on the fit from Eqs. (25) Figure 6.12 Ratio of the normalized fatigue -equivalent bending strain from the VIV tests divided by that calculated by applying F105 [ 3] with arch action as described i n Appendix D. 7 Structural Damping The VIV test models [19,22,23] were designed to ensure low structural damping. In view of this and the great uncertainty about the structural damping for the modes, frequencies and amplitudes excited, the structural damping has not been included in the analysis of the test data. Structural damping tests have been performed in air , since even in still water hydrodynamic damping is significant . The pipe is manually excit ed at the quarter point (25% of the span length from one end), and the decay in response after the excitation ceases is used to estimate the damping ratio. Thus, f or the OL tests, around 0.4% damping is measured for Modes 1 to 3; for the EM tests, “the structural damping was meas ured to be less than 0.3% of critical” [ 19]. For the Shell -sponsored tests on Pipes 1 to 3 , structural damping values are not reported in [ 23], but [ 13] does report an “average” structural damping ratio of 0.5% for Pipe 1. 25 For Pipes 2 and 3 , re-examining the acceleration records during the damping tests, filtered to include the frequency range 1 to 5 Hz only, and double integrated to co nvert accelerations to displacement s yielded the following: For Pipe 2 the damping ratios range from 0.6% to 0.8% for Mode 2 at amplitudes ranging from 5.1 D to 2.9 D . For Pipe 3, 1.5% damping is obtained at amplitudes ranging from 2.8 D to 0.7 D. Modes 2 and 3 were excited which result ed in some beating in the response , which made it more difficult to accurately estimate damping from the decay in amplitude of free vibrations. It is thought that the higher damping for Pipe 3 compared to Pipe 2 may be due to the half shells used to increase the hydrodynamic diameter of the pipe. All these damping tests involve low modes, whereas the VIV response often involved much higher modes . (L̃ is essentially the mode number that is excited at the Strouhal frequency.) Thus, there is great uncertainty about the actual structural dampi ng for the modes, amplitudes, and frequencies of VIV, as noted in [ 13]. Although models typically include viscous damping, for which the damping force is proportion al to the velocity, the damping could also be hysteretic (with a magnitude of the damping force that does not change with velocity), or quadratic. This affects how the energy dissipated per cycle varies with amplitude and frequency. E.g., (Coulomb) frictional dissipation is more likely to give ris e to hysteretic damping for which the energy dissipated does not change with frequency and is proportional to the amplitude itself , whereas for viscous damping the dissipated energy per cycle is proportional to frequency and amplitude squared. Moreover, the damping mechanism is unclear. In [ 13] the structural damping is modeled in the same way as hydrodynamic damping, by dis tributed dashpots applying a transverse force on the pipe that opposes the transverse component of pipe velocity. This may be realistic for soil damping where the pipe is supported by the soil, but not for the structural damping of the model tests. A more reasonable approach is to include structural damping where there is structural stiffness. Three categories of structural stiffness and the associated damping can be distinguished: (1) that associated with arch action, (2) that associated with the axial tension without arch action , and (3) that associated with flexural resistance . The first could be important, f or low modes, e.g. affecting the measured damping value for Mode 1, but since the higher modes are hardly affected by arch action, the same will apply to any damping associated with arch action (i.e. damping in the axial deformations of the pipe, in the sp ring that maintains the tension, and in any of the associated connections and hinges). The second is zero at amplitudes sufficiently small that the linearized theory applies [14]. Only the last has been included in the formulation (Eq. (5)). Structural nonlinearities can also affect energy dissipation, e.g. by changing the frequen cy of part of the kinetic energy to one that is more strongly damped. Assuming that the measured damping ratios are due to flexural, viscous damping, the damping coefficient C k can be calculated, and used to calculate the normalized damping values C̃k for each VIV test. However, this gives normalized damping values C̃k that are far too high for some of the VIV tests. Thus, flexural viscous damping cannot be the only source of damping in the damping tests. Assuming flexural hysteretic damping would lead to much lower C̃k values. In theory, whether the damping is viscous or hysteretic could be determined from be determined from how the decay rate of the vibrations during the damping tests depends on the amplitude, but re -processing the damping test acceleration records for Pipes 2 and 3 did not reveal a clear amplitude dependence of the decay rate. Thus, there remains a large uncertainty about the amount of damping present in the VIV tests, but it is not expected that the structural damping is sufficiently high to have a large effect of on cross -flow VIV. Nevertheless , it is conceivable that for the extremely high -mode response of Pipe 1 in Figure 6.8, the downward trend with increasing L̃ for L̃ > 20 may in part be due to structural damping, possibly associated with geometric nonlinearity for such high -mode response. The authors fully concur with the recommendation in [13] that more attention should be focused on understanding and quantifying the structural damping of VIV test models for the modes excited by the VIV. For insta nce, if necessary by also performing damping tests on shorter span models, and performing tests with accurately -controlled harmonic excitation as well as free vibration decay tests. 26 8 Prototype Example 8.1 Prototype Definition Here “prototype” refers to a pipeli ne span to be assessed. For the example considered, the span length is L=150m; the steel pipe outer diameter is 24inches (0.6096m); the wall thickness of steel is 17mm, the corrosion coating has a thickness of 6mm and density of 1300kg/m3, outer concrete weight coating thickness is 40mm with a density (including water absorption) of 2400kg/m3, the steel has Young’s modulus of 207GPa, and a density of 7850kg/m3, and the densities of the gas inside the pipeline and the seawater outside are 100kg/m3 and 1025k g/m3, respectively. The stiffness of the coatings is neglected. The mass ratio for the pipe is thus calculated to be c pipe = 1.234. To match the model tests, the ends are assumed to be simply supported, and the level of axial restraint is determined by a ssuming linear distributed axial soil springs with a stiffness of 5MN/m2, which implies L ea = 2268m, and an equivalent axial spring stiffness at one end of the span of k x = 90.5 MN/m. (Use of the equivalent end spring allows the same formulation given in Appendix B for arch action to be used for this prototype span as it is for the model spans .) The effective axial force under still -water conditions is assumed to be 1MN. However, this axial force increases with the flow, as determined by the static solution of Appendix B using a drag coefficient of cd = 1.5. (An alternative might be to use the drag coefficients from Figure 6.5 determined for the model tests from the measured change in axial force due to the flow. Those account for the increase in drag due to VIV, but may suffer inaccuracies e.g. due to force measurement errors, unreco rded thermal expansion, or creep, especially where the increase in axial force due to the flow is small.) For the fatigue resistance, the API X’ curve is used for which the exponent is n = 3.74, and afat = 6.89 × 1035 Pan. (The same value of the exponent is used for the model tests to evaluate the normalized fatigue -equivalent strains, as it should be. The value of a fat is not needed to calculate any of the parameters reported for the model tests, but it is needed to report the actual fatigue damage rates for the prototype.) The above bears some similarity to a span assessed by some of the authors in which the pipe crossed a shallow channel. For a limited time, a turbidity current (“TC”) travelling down channel can reach high current velocities. The late ral displacements it produces increase s the axial tension and thus the span length. However, the example here has been much simplified, to focus on issues that can be captured by the model tests, enable the use of the analytical solution of Appendix B, an d ensure easy reproducibility of the results. Phenomena for the real span under TC loading that are not model led include: pipe soil interaction at the shoulders, including nonlinearity loss of contact and energy dissipation, nonlinearity and history depe ndence in the axial resistance at the shoulders, and the increase in span length with increasing current velocities and axial forces. Although, these phenomena are certainly important, and the scaling theory presented here could be used to design model te sts to address them, they are beyond the scope of this paper. 8.2 Dominant Half Wavelength, λ To apply scaling Assumption ( 9), the dominant half wavelength, λ, is needed. This is done in different ways according that what data are collected for the tests: For the Shell tests (Pipes 1 to 3) acceleration data collected at 22 points along the span are used to perform Fourier anal ysis. For the other data a dominant mode number, n dominant , is estimated by inspecting the data (using engineering judgment), and the dominant wavelength is then calculated from λ = L/n dominant . The resulting dominant wavelengths are shown in Figure 8.1 in normalized form as λ̃ = λ/Lst. 27 Figure 8.1 Normalized dominant half wavelengths, λ̃ = λ/L for model tests 8.3 Scaling R esults First the static solution is calculated, this then provides the required inputs for scaling the dynamic response from the model tests. The former is done analytically, with the result from Appendix B. For that purpose it is easiest to assume a val ue of the axial force N under flowing conditions, and then determine the current velocity U c at which that axial force is reached. The resulting static solution is shown i n Figure 8.2 to Figure 8.4. Figure 8.2 Effective axial force from static solution for prototype span 28 Figure 8.3 Midspan deflections from static solution for prototype span Figure 8.4 End rotations (in radians) from static solution for prototype span , combined rotation from gravity and hydrodynamic drag force. The scaling results are applied to the dynamic part of the response , using the extended scaling approximations of Section 5 as well as the basic assumptions of Sectio n 2. Thus, for every model test , a prototype current velocity is determined for which similarity is satisfied. The steps involved are as follows: 1) Start with an assu med axial force N and calculate the corresponding static solution including the prototype current velocity U using the static solution of Appendix B. 2) Determine the dominant half wavelength λ and the Strouhal length L st for the prototype by applying the scale factor (i.e. prototype/model ratio of span lengths) to the model values. 3) Calculate adjusted values of the flexural rigidity and effective axial force for the prototype, EI adj and N adj from Eqs. (20) and (21), using the model values for the normalized quantities, and the prototype values otherwis e. 29 4) Repeat the above steps updating the assumed N in Step 1 until Eq. (8) is satisfied with the adjusted values of EI and N, and with T st calculated for the prototype usi ng Eq. (7). From the static solution this then also gives the prototype current velocity U for which similarity with the model test is satisfied. Here the solution is sought for prototype axial forces in the range 1.0000000001 MN to 22.28MN, corresponding to prototype current velocities in the range 0.005m/s to 10m/s. If no solution is found in that range, the model test is considered not relevant. For such cases i t may represent even higher current velocities , but the axial force corresponding to 10m/s current velocity is already more than the pipe could sustain without yielding. 5) Any parameters of interest may now be calculated by multiplying the normalized values from the model test by the unit value for the prototype. For instance the unit value for the fatigue damage rate are calculated with Eqs. (17) and (14), which may be combined to yield η̇1=[ RϵD/(L st2ϵfr)]n/Tst, or, if the fatigue resistance is charac terized by a fat rather than ϵfr, η̇1=[E RϵD/L st2]n/(afatTst). The fatigue damage rate then becomes η̇ = η̃̇ η̇1 = (2ϵ̃e)n η̇1 =[2ϵ̃e E RϵD/L st2]n/(afatTst). These fatigue damage rates are plotted in Figure 8.5 for the relevant model tests together with the relevant current velocity. The above algorithm can easily be adapted if the static solution is determined by FE analysis rather than analytically . The FE analysis is then performed for increasing values of the current velocity, until conditions are such that Eq. (8) is satisfied, as in Step 4 of the above algori thm. The static FE analysis may then include any desired nonlinearities, including changes in span length, L. Figure 8.5 Prototype fatigue damage rates as a function of current velocity from scaling of model tests, and by applying F105 [ 3] with and without considering arch action in the modal analysis 8.4 Predictions from DNVGL Recommended Practice F105 For c omparison with the scaled model test results, the predictions from F105 [ 3] are also included in Figure 8.5. For that purpose the modes are calculated with and without including arch action, using the analytical solution of Appendix B. Further details needed in addition to [ 3] to make the F105 calculation results reproducible are given in Appendix D. For low current velocity , F105 predicts zero response and fatigue damage rate, which is not included in Figure 8.5, because it would be off the logarithmic scale used. 8.5 Vivana Analyses For comparison the prototype pipe is also analyzed with the frequency -domain version of Vivana [ 4]. The structural model in VIVANA consists of 300 elements of equal length. Both ends are pinned except for that the pipe is free to extend in the axial direction a t one end, where a n axial force is applied 30 that is a function of the current velocity only, as given in Figure 8.2. Thus, arch action is neglected. The default hydrodyn amic lift and mass coefficients are used, which are generalized based on labora tory test data [ 4]. The drag coefficient is taken to be 1.5, as for the F105 analyses. The Vivana static lateral and vertical deflections are indistinguishable on a plot from those in Figure 8.3. Pure CF analysis yields the fatigue damage rates include d in Figure 8.5. Although the Vivana CF fatigue damage rates tend to increase with current velocity, as expected, this increase is not monotonic. The sudden drops app ear to be associated with mode shifts, i.e., the responses reduce before the next mode can be fully excited. 8.6 Partial Strake Coverage Some tests on Pipe 3 include strake s with a gap in the strake coverage centered at midspan. E.g. the span is “75% straked” if the gap in the strake coverage at midspan is 25% of the span length. The strakes are described in [ 23]: they have a he ight of 0.25D h and a pitch of 15D h, with 3 starts so that there is one strake at every 120° of the circumference. Here D h = 1.075 D is the hub diameter, i.e. diameter of the pipe from which the strake fins protrude . (For Pipe 3 the outer diameter of the bare pipe is D=80mm, but adding the half shells with strakes the hub diameter is D h=86mm.) The analysis to determine the fatigue damage rate from the model tests for a prototype with similar strake coverage is the same as for the bare pipe (as per Section 4), except that the effect of the strakes on the drag coefficient should be considered in calculating the static solution for the prototype. The strakes tend to increase the drag, but so does VIV which mainly occurs when no strakes are present. For simplicity the same drag coefficient of 1.5 for the bare prototype is also used for the straked parts of the prototype . This is close to the drag coefficients of around 1.4 reported in [ 24] for pipes with similar strakes, except that the pitch of the strakes in [ 24] is 17.5 D instead of 15 D. (An alternative is to use the effective drag coefficients determined indirectly, as in Appendix B, from the VIV tests on the straked pipe, assuming the drag coeffi cient is constant along the length of the pipe.) The resulting cross flow fatigue damage rates for the prototype straked pipe are shown in Figure 8.6. As expected, the strakes bring about a large reduction in fatigue damage rate, though some of the 48% straked Pipe 3 tests show a higher fatigue damage rate than EM tests on bare pipe corresponding similar prototype current velocities. The in line fatigue damage rates fo r the partially straked pipes are not shown in the figure, but are all lower than the corresponding cross flow ones. Figure 8.6 Prototype cross flow fatigue damage rates for partially strake covered pipe sh own as solid squares together with the cross flow data for bare pipe from Figure 8.5 8.7 Discussion of Prototype Results If all scaling approximations are good, the scaled experimental results in Figure 8.5 should coalesce into a single line. That relies on the extended as well as the ba sic scaling approximations . Clearly the 31 scatter about this “single line” is large. To quantify the scatter, the F105 predictions are used as a reference line that reasonably captures the velocity dependence of the fatigue damage rate. Considering the da ta in Figure 8.5 for current velocities in the range 0.6 to 10m/s, the scaled/F105 ratio of fatigue damage rate s range from 0.0408 to 47.7, with a geometric mean value o f 0.72 and a logarithmic standard deviation of 1.80. For in line, this scaled/F105 ratio of fatigue damage rates ranges from 0.0376 to 6.78, with a geometric mean value of 0.35, and a logarithmic standard deviation of 1.54. It is not clear that causes this high scatter in results , but, whatever it is , is not captured by the F105 analysis . Neglecting the mass ratio c pipe (as per extended scaling Approximation ( 8)) could well be responsible for the relatively low fatigue damage rates for the ExxonMobil (“EM”) tests, with cpipe = 2.22 , compared to Pipe 3 with cpipe = 1.14 . These correspond to the highes t and lowest values of c pipe, respectively; s ee Table 6.1 for the mass ratios of the other tests. Other test data that show a similar trend with respect to the mass ra tio, include the 1 or 2 degree of freedom tests of a rigid cylinder supported by springs in [18], which cover a very wide range of mass ratios . There the mass ratio hardly affects the peak response , but it has much more effect on the range of current velocities over which high response occurs. In the multi -mode case, this could translate to more modes responding simultaneously. Adjusting N and EI to match the model Ñ values as per Eqs. (20) and (21) achieves the correct structural stiffnes s to resist pipe out of straightness at the dominant wavelength, but not at other wavelengths. To assess the error of the approximation, the stiffness at half the dominant wavelength is considered. In particular, the ratio of this stiffness after the adj ustment divided by that before the adjustment is plotted in Figure 8.7. It ranges from 0.56 to 1.4. There is no apparent correlation between the scatter in Figure 8.5 and this stiffness ratio . This does not necessarily mean that the error due to adjustment of EI and N is insignificant, but the test data used appears insufficient to quantify what it may be. Figure 8.7 Ratio of structural stiffnesses to resist out -of-straightness at half the dominant wavelength divided by the unadjusted value 9 Conclusions Advantages of the scaling developed include: 1) The number of dimensionless parameters is reduced by scaling assumptions that serve as approximations . These include local action by which D/L need not match, and using the Von Kárm án beam theory for moderately large deflection s with constant axial force to reduce the number of parameters required for the “complete” [ 10] structural characterization of the span . 32 2) Using the Strouhal period T st and half wavelength L st as the units of time and length in the axial direction, and the hydraulic diameter D as the unit of length in the transverse direction makes the normalized parameter s physically meaningful . It also mak es it clear how to extend the scaling, e.g. to include a soil -supported part of the pipe. 3) The normalized parameters are stable ; they do not vary widely with the normalized VIV test design parameters . 4) The normalized fatigue -equivalent strain amplitude ϵ̃e is not only stable, but also gives the fatigue damage rate directly, based on the rainflow counting performed for the test data. The basic scaling approximations applied to a simply supported structurally undamped pipe lead to ϵ̃e=ϵ̃e(L̃,Ñ,cpipe). I.e. that the normalized fatigue -equivalent bending strain is a function of the normalized span length, the normalized effective axial force, and the mass ratio. Further, extended scaling approximations in principle allow any VIV test to be scaled to any prototype, provided the end conditions are similar (e.g. both simply supported) , and the model and protype current velocities are related for similarity. To assess a prototype span, the static equilibrium configuration is determined by analysis. Thi s completes the input to the scaling of the dynamic response. This scaling is applied to VIV test data from well -instrumented tests from three testing programs carried out at Marintek (currently SINTEF Ocean). The data do not allow a direct experimental verification of the scaling approximations , because there are no two tests on different pipes for which the normalized input parameters (L̃,Ñ,cpipe) are the same. However, the data do allow comparisons of different test programs , and provide data points f or estimating the function ϵ̃e=ϵ̃e(L̃,Ñ,cpipe) for essentially undamped, simply -supported spans. The scatter in the results suggest that either the function ϵ̃e=ϵ̃e(L̃,Ñ,cpipe) is not smooth, or inaccuracies in the scaling approximations are considerable . The latter are assessed (in Appendix C) using relative error indicators denoted by ψ , and the data used have been selected to exclude those for which larger scaling errors are incurred. Where clump weights have been used to stabilize the ends of the tes t pipe, axial modes at around the frequency of VIV response may be excited by geometrically nonlinear affects, including those that are nonlinear in the dynamic increment, and those termed “arch action” arising from the static loads (i.e. that the static l oads change the behavior for small superpose d increments). 10 Acknowledgements The authors are grateful for permission to use the data from tests sponsored by ExxonMobil, Shell, and the Ormen Lange License Holders (Petoro AS, Equinor Energy AS, A/S Norske She ll, PGNiG Upstream Norway AS, and Vår Energi ASA). This research did not receive any specific grant from funding agencies in the public, commercial, or not -for-profit sectors. 11