Root mean square force coefficients for submarine pipelines

Ocean Engng, Vol. 15, No 1, pp. 55--69, 1988. Printed in Great Britain. 0029-8018/88 $3.00 + .00 © 1988 Pergamon Journals Ltd. ROOT MEAN S Q U A R E FORCE COEFFICIENTS FOR S U B M A R I N E PIPELINES N. JOTHI SHANKAR, HIN-FATT CHEONG a n d K. SUBBIAH Department of Civil Engineering, National University of Singapore, Singapore 0511 Abstract--An experimental investigation has been carried out on forces induced by regular and random waves on submarine pipelines placed near a plane boundary. The inline and transverse forces are analysed in terms of combined root mean square (rms) hydrodynamic coefficient. The total rms coefficient is correlated with Keulegan-Carpenter number or period parameter and relative clearance of the pipeline from the plane boundary. The pipeline was subjected to Pierson-Moskowitz spectrum (P-M spectrum) in the random wave force tests. The time histories of water particle kinematics are generated using the linear numerical transforms. This paper also reports the effect of depth parameter on the total rms coefficient. The results of the random wave force tests are finally compared with those of regular waves. NOMENCLATURE CD c~ CFHrms CLmax' CLrms CTrms D d e fnrms Fwms H K k L Re T l Umax Urms Wrms U W Z f~ p P fl Drag coefficient Inertia coefficient Inline rms coefficient Maximum and rms transverse force coefficients, respectively Total rms force coefficient Diameter of the pipe Still water depth Vertical clearance of pipeline from the bottom boundary Inline rms force per unit length of pipeline Transverse rms force per unit length of pipeline Wave height Keulegan--Carpenter number or period parameter Wave number (= 2~r/L) Wave length Reynolds number Wave period Time variable Amplitude of inline water particle velocity Inline rms water particle velocity Transverse rms water particle velocity Inline water particle velocity Transverse water particle velocity Depth of submergence of pipe axis from still water level Frequency parameter Kinematic viscosity of water Mass density of water Velocity ratio. 55 56 N. JOllll StlANKAR, HIN-FAII ClIEONG and K. SUBBIAIt 1. INTRODUCTION THE EVALUATION of hydrodynamic forces is a vital aspect in the design of offshore structural elements. The Morison equation (1950) is the most commonly used semiempirical model to compute the wave-induced forces on slender structural members whose lineal dimensions are small in relation to the wave length, L ( D / L < 0.2; D is the diameter of the structure). The Morison equation is a linear summation of drag and inertia forces, in which the former depends on the water particle velocity and proportional to coefficient of drag, CD and the later depends on the water particle acceleration and porportional to coefficient of inertia, C,,. These hydrodynamic coefficients vary with flow conditions around the structure, water depth and presence of free surface and plane boundary. Studies on the wave forces on submerged cylinders have been carried out by many investigators. Nath and Yamamoto (1974) found that C,,, for a cylinder placed on the boundary is more than twice for a cylinder in a free stream. Yamamoto et al. (1974) concluded that using potential flow theory, the coefficients can be properly evaluated if the wake dependent drag forces are negligible. Nath et al. (1976) reported that the wake conditions have significant influence on hydrodynamic forces. Sarpkaya (1975) from his planar flow experiments found that in the transverse forces several frequencies may occur in a given cycle. It is also reported by Sarpkaya (1976, 1977) that the vortex shedding frequency is not a pure multiple of flow oscillation frequency and the number of vortices shed in each cycle increases with increasing period parameter, K (= U......T/D; U ...... is the amplitude of inline water particle velocity. T is the wave period and D is the diameter of the cylinder). A classification of various flow conditions during a wave cycle around a submerged cylinder has been discussed by Wright and Yamamoto (1979). They found that the increase in water depth generally produced increase in Cm and lift coefficient C t . Ramberg and Niedzwecki (1982) cautioned that the hydrodynamic coefficients obtained from planar flow results cannot be used to evaluate the hydrodynamic forces except possibly at larger values of K. The existing literature on the hydrodynamic coefficients indicates a large scatter in the values of C n than C,,, when correlated with K. However, it has been found that the presentation of wave forces as a single root mean square (rms) coefficient considerably reduces the scatter and avoids any phase problems. Maull and Milliner (1978) concluded from planar flow experiments that the inline rms force could be evaluated using Morison equation with C,,, equal to 2 and another component as a function of motion of the vortices. Maull and Norman (1975) found that the Morison equation overpredicts the forces when the coefficients are taken from planar flow experiments for large values of relative clearance of pipeline, e/D (e is the clearance between pipe and plane boundary) but for smaller values of e/D. the prediction of Morison equation is good. Ramberg and Niedzwecki (1982) observed that their results over predicted Maull and Norman's rms coefficients but found that as the velocity ratio, Y~ (= W, ..... / U ...... ; W,..... and U...... are amplitudes of transverse and inline water particle velocities respectively), was decreased, the rms forces tended toward the planar flow results. Stansby et al. (1983) reported that when ~q is < 0.7, the rms forces were close to those of planar flow results. It was found by Chakrabarti (1985) that when ~ was between 0.3 and 0.9, the inline rms coefficient was insensitive to K between 0 and 40. Bearman et al. (1985) RMS Coefficients for submarine pipelines 57 attributed the difference between measured and computed forces using Morison equation due to strong vortex shedding. This paper presents the results of the experimental investigation on wave-induced forces due to regular and random waves on submarine pipelines placed near a plane boundary. The total rms coefficient resulting from the combination of inline and transverse forces is correlated with period parameter or Keulegan-Carpenter number and relative clearance of the pipeline from the plane boundary. The pipeline was subjected to P-M spectrum and the time histories of water particle kinematics are generated through the use of linear random wave model. This paper in addition to investigating the effects of depth parameter on the total rms coefficient compares the random wave force results with those of regular waves. 2. T H E O R E T I C A L C O N S I D E R A T I O N S Figure 1 shows the definition sketch of the wave-pipeline interaction problem. Let a train of regular small amplitude surface waves of height H and period, T propagate past a submerged pipeline fixed at a clearance, e, above the plane boundary and induce hydrodynamic forces on the pipeline. Let d be the still water depth and F , and Fv be the two component forces acting in the inline and transverse directions respectively. Oirection of ;~ L z wave propagation [ ""_.L._ _I I ~. 1 f "-T z:O pipeline--~~_~ ...) ......... e .... z : - f l . . Piano.boundary FIG. 1. Definition sketch. The root mean square (rms) force, Frm.~of a force trace is computed as, rrm~ = [lf7-1.FZ(t) dt]]o.5 (1) in which F is the force at time t. Thus the inline and transverse rms coefficients, CFmm~and Ct_rm~ defined in terms of the amplitude of inline water particle velocity, Umax are given respectively as, FHrms CFm,-,~= 0.5pOVZmax (2) Fvrms CL~m~- 0.5pDU2max (3) 58 N, JOIHI SHANKAR, H]N-FATr CHEONG and K. Su~m,xH in which F H r m s and Fvrms a r e the inline and transverse rms forces per unit length of the pipeline, p is the mass density of the fluid and the other parameters have been defined earlier. The total rms coefficient, CTrms combining the inline and transverse forces and normalized in terms of inline and transverse rms water particle velocities, Urm~ and Wrms respectively may be defined as, .... ],,2 [ F 2. .... + CTrms ~ 0.5pO[U2ms + Wr2ms] (4) The inline and transverse water particle velocities, u and w can be written respectively in terms of Uma× using small amplitude wave theory as, u = UmaxCOS0 (5) W = -- Um,x tanh k(d+z)sinO (6) in which k is the wave number (= 2~r/L; L is the wave length), z is the depth of submergence of the pipeline from the still water surface being measured positive upwards from still water level and 0 is the wave phase angle. The rms velocities Urm~ and W r m s c a n be derived using simple integral calculations as, Urms - Wrm s = Vmax ,- (7) \/2 Umaxtanh k(d+ z) ¢- (8) \J2 Substituting Equations 2, 3, 7 and 8 and z = - d + (e+0.5D) in Equation 4, Crrm~ can be deduced as, CTrms = 2[C2 rms + C2rms],/2 (9) {1 + tanhZ[kD(0.5 + D)]} 3. EXPERIMENTAL INVESTIGATIONS The present experimental study has been carried out in a wave flume measuring 36 m in length, 1.3 m in height and a uniform width of 2.0 m. The Danish Hydraulic Institute (DHI) paddle type wave generator installed at one end of the flume can generate regular and random waves according to the input signal and a long beach at the farther end absorbs the incoming waves so that reflection is very much minimal. The test section was chosen near the middle of the flume. Force d y n a m o m e t e r s based on strain gauge principle, were used to measure the total instantaneous inline and transverse forces on a 60-mm diam. smooth perspex pipeline model. These force dynamometers were essentially of cantilever beams, carrying a full bridged form of strain gauge arrangement. These cantilever beams were fixed to the bulk heads and the whole arrangement was embedded rigidly inside the model at its ends. The force dynamometers were aligned orthogonal to each other so that the one in the horizontal plane measured the total transverse forces while the other in the RMS Coefficientsfor submarine pipelines 59 vertical plane measured the total inline forces. The ends of the pipeline were properly sealed to prevent entry of water into the model. The model was supported at the ends by self-aligning bearings. The instantaneous water surface elevation above the axis of the model was measured using the resistance type wave gauge. The pipeline model was subjected to various regular waves of heights and periods ranging between 45 and 300 mm and 1.0 and 2.5 sec, respectively. The experiments were carried out for various e/D values of 0, 0.1, 0.25, 0.5, 0.75, 1.0 and 1.5. Three different water depths of 0.54, 0.48 and 0.42 m were used in the experiments. The model was subjected to P-M spectrum at various energy levels in the random wave force tests so as to achieve various significant wave conditions. The same e/D values and water depths as in the regular wave experiments were used in the random wave force tests also. The time histories of instantaneous water surface elevation and the corresponding total inline and transverse forces were recorded using a HP data acquisition system. The number of data samples in the regular wave force tests was 512 while in the random wave force tests it was 1024 per realization. The data were acquired at the sampling rate of 15,000 ixsec/channel and stored in diskettes for further analysis. The detailed experimental procedure and data acquisition system have been reported earlier by Jothi Shankar et al. (1986). This experimental study covers a Reynolds number range of 3000-38,000 and period parameter of 0.5 to 28 both in the regular and random wave force experiments. 4. DISCUSSION OF RESULTS Using dimensional analysis it can be derived that any force transfer coefficient, CF is in general a function of fluid and flow parameters and pipeline conditions. CF = f(e/O, d/D, K, Re or 13) (10) in which d/D = depth parameter, Re = Reynolds number (= Um,xD/v; v is the kinematic viscosity of the fluid), 13 = frequency parameter (= RJK) and other parameters have been defined earlier. Sarpkaya (1977) has reported that either Re or 13 can be used as an independent parameter to correlate the hydrodynamic coefficients since v appears in both these parameters. The variation of Crrm~ with K for e/D = 0.25 and d/D = 8.0 and various 13 values ranging between 1440 and 3980 is shown in Fig. 2. Though this typical plot indicates a definite trend that Crrm~ decreases with increase in [3 for a given K as obtained by Sarpkaya for the variation of Cm with 13, no distinct variation in the overall magnitudes of the coefficients is observed at least within the range of 13parameter encountered in the present investigation. Similar results were also obtained by the authors (Cheong et al., 1987) with regard to the variation of inline inertia coefficient, C,,, with 13. It is found from the present investigations and also from Sarpkaya (1977) that R e is not an effective independent parameter to correlate the hydrodynamic coefficients at least within the present range of Re = 3000 to 38,000 and hence further correlation is attempted mainly with K for the overall variation of the hydrodynamic coefficients without regard for 13 values. 60 N. Jo'rH[ SHANKAR. HIN-FAFI CHEONG and K. SummxH 0 e/P :: 0 2 5 J d/D = 80 m Legend a~ n E~0 J • 16 =3980 o ,..) n ~.._~A?.Q... v±<< o • ~__.g7_<o._ < n ~:~ =e35o__ n /5' = , C O 0 0 7 : • #_ : 1 8 0 0 J ~ : , 4.4.0. _ ~7 I I d 4 i I I i I I I ~ lO L2 i4 [b [8 7 20 FiG. 2. Variation of C/ ..... with K and 13. Behaviour of CT. . . . . . with d/D The correlation of CTrms as defined in Equation 4 with K is depicted in Fig. 3 for various e/D and d/D values. It is observed from these plots that CTrms decreases rapidly with increase in K for lower values of K (K < 5) for all the d/D and e/D values. As K increases further, CTr,,s decreases gradually. It is also noted that there exists a local increase in the value of Crrm~ which decreases with decrease in d/D and is shifted towards higher values of K. These effects can be attributed to the inclusion of transverse hydrodynamic components in the definition of Equation 4. The authors (Cheong et al., 1987) have already reported that the transverse hydrodynamic coefficients, CL.... (maximum transverse force coefficient) and CLrms decrease with decrease in d/D for given values of e/D and K. They also concluded that the peak values of these two hydrodynamic coefficients indicate a shift towards higher K values along with the decrease in magnitude with the decrease in d/D. It is a well known fact that the frequency of vortex shedding and the transverse force increase with increase in K (Sarpkaya, 1976). For low values of K, (K < 5) the flow around the pipeline does not separate and the transverse force, which oscillates at the same frequency as the wave, is due to the asymmetry of pressure distribution around the cylinder. As K increases, K > 5, vortices will be generated but will not move away from the pipe as the wave will be reversed in every half cycle and the water particles will not have sufficient time to travel to a long distance as to move the vortices away from the pipe. In this range of K, the transverse force will no longer oscillate at the same frequency as the wave but increases with K. This transient flow phenomenon which occurs for values of K around 5 and 15 causes a local increase in the value of CT. . . . depending on d/D and then CTrm~ decreases gradually with further increase in K. The combined effect of velocity and acceleration and the resulting pressure and shear RMS Coefficients for submarine pipelines 61 50 60 ~_ e/D = 0 0 10 ~o c) +~+'~+ ~-~ o+ ,+ I I 5 ~'~' I I0 I 5 20 15 I I 15 10 I I 20 25 30 50 50 %% e/D = 0 2 5 e/D = 0 5 ~%• ~1,410 (9 t L I I I I 5 i0 15 20 + ). I 25 50 Io ~ , e/D = 0.75 lo t-- I I c) ~ , 0 I 5 I 10 I 15 I+ 20 0.3 25 I+ 30 I 20 25 30 e/D = 1.0 "-1: +~.++ 1 03 I 0 5 10 15 50, 30 I I I I 5 10 15 20 + I+ 25 30 K 1oI K 50 e/D = Legend 1.5+ 1 • = 1440 - 3980 • L I I I I I 5 10 15 20 25 = 9.0 d / D = 8.0 '"~'. +"~ • 03 / 0 d/D d/D = 70 30 K FIG. 3. Variation of Crrm, with K for various e/D. distributions around the pipe cause the force coefficient to decrease. These plots also indicate that Crrms decreases with decrease in d/D for any given value of e/D and K. It is observed clearly that CTrmsdecreases with decrease in d/D even for the small range of d/D investigated. This can be partly attributed to the nonuniform flow conditions around the pipe (Wright and Yamamoto, 1979). The mechanics of the flow phenomenon around the submerged pipeline near a plane bottom boundary varies greatly with water depth, proximity of the pipe to the bottom boundary and wave conditions. Since the 62 N. JoTm SHANKAR, HIN-FAI'] CHEONG a n d K. SUBBIAI! flow mechanism around the pipe is quite complex due to these factors, it is believed that the variation of Crams with d/D for a given K could be better explained by detailed measurements of pressure distribution around the pipe, body distorted water particle kinematics near the cylinder and also in the bottom boundary layer. Behaviour of CTrms with e/D Figure 4 shows the variation of CTrmswith K for various d/D and e/D values. It can be seen in this figure that Crrms decreases with increase in e/D for given values of d/D and K. This may be attributed to the effect of boundary proximity, the effect of which gets reduced as the pipe is moved away from the boundary. As in the case of variation of Crrms with d/D, it is also observed in these plots that as K increases between 5 and 15, CTrmsincreases locally depending on the value of e/D and then decreases gradually with further increase in K. It is further seen that the local increase in the peak value of Crrms decreases with increase in e/D and is shifted towards larger values of K. This type of variation for CT~ms is noticed for lower values of e/D (< 0.75) indicating the effect of boundary proximity on the transverse forces. However, it is further noted that the trend of CTr,,,, with a decrease in peak value and its shift with increase in e/D is not noticeable for higher values of e/D indicating that the lift phenomenon gets reduced as the pipe is moved away from the boundary. 75 :! 75 •w•<•%jd/D [,.., d/D = 0 = dl 80 L) L) ,.7 ~ , ~ ... 7 ?,%~ "%~% t I I I 5 10 15 05 20 0 I I I 5 [o t5 K o< > 7 7 K =t440 -3980 75 Legend 1o ~b. d/D 70 • e/D = 0 0 = e/D=OL b.-, e/D = 0 25 o e/D:05 e/D = 0 75 0~ I I I I I 5 i0 15 20 25 o 30 7 K FIo. 4. V a r i a t i o n of CrT.,.~ with K for various d/D. e/D : 1.0 e/D = 1.5 . × 17 20 25 RMS Coefficients for submarine pipelines 63 Comparison of measured and computed CTrms The comparison between the measured and computed CTrmsas in the semi empirical Equation 9 is depicted in Fig. 5 for two e/D values as typical plots. These plots show that measured Crrmscorrelates very well with the computed CTrms. However, it is noted that almost all the data of computed Crrms fall slightly below the 1:1 correlation line, indicating that the Crrms as defined in Equation 9 slightly underpredicts the total rms coefficient. This small difference may be attributed to the variation of the vortex 75 = o.2. J'j+ o ,~0 10 Legend [... L) * d/b = 9.0 / / ~ d/D= flO + I I i i I I I I I I d/D 7.0 = I ' ' CTrms ' ' 75 tO (rneas) 50 e/D = 10 I0 0 Legend k, E'- (J • f d / D = 9.0 ~ d/D = 8.0 • d/D = 7.0 03 0 I0 I CTrms (meas) FIG. 5. Comparison of measured and computed Cr~m~. 50 64 N. JoTm ,~I|ANKAR, HIN-F'Arr ('m~ONC, and K. SUBI31AH shedding frequency with K and may partly be due to the usage of computed kinematics instead of the measured kinematics at the centreline of the pipe. Despite this small difference, the comparison between the measured and computed CTrm~may be regarded as very satisfactory. Comparison of CTrms with CFrl ...... It has been observed from the present data, that the inline runs coefficient, CFH.... as defined in Equation 2 is close to Crrm.~. Figures 6a and b show the typical plots for the variation of CFHrms and Crrm~ with K for all the d/D and e/D = 0.25 and 1.0, respectively. These plots also depict the comparison of inline and total rms coefficients. For the sake of completeness, the transverse rms coefficient, Cr . . . . given by Equation 3 is also superimposed. It is observed that CTrms is almost coinciding with the corresponding CFHrm s for all the values of K encountered in the study. This may be due to the fact that CL,-msis small within this range of K. The purpose of superimposing CLrm~ on these plots is to show that CT~m~exhibits a local increase in its value for values of K around 5 and 15 which is due to the transverse force component. From the above discussions, it is clear that Crrm~ and CFHr,,~ are almost coinciding with each other at least within the range of K in the present study. In order to ascertain, the extent of difference between these two coefficients, typical plots were drawn between ACorns and K for e/D = 0.25 and 1.0 in Figs 7 a and b, respectively. The AC, m, is defined as, ACrrns • CTrms--~--CF-Hrm-'-s (1 l) CTrm s These plots indicate that AC~m~ decreases with increase in K but is seen to be independent of d/D. The average absolute relative difference is found to be about 7.5% at very low values of K. Effect of kD o n CTrms The equation that is derived for CT~r~ (Equation 9) is found to he a function of scattering parameter, kD. In order to determine the effect of this parameter on CT~.... sample plots are drawn showing the variation of Crrm~ with K for different values of kD, e/D = 0.25 and all the d/D investigated in Figs 8 a-c. These plots do not show any trend of Crr,,,s with kD for any given values of d/D and K which is as expected. It is well known that the scattering will be dominant only when the linear dimension of the structure is such that the presence of it does affect the incident wave field. As the diameter of the pipe is very small in relation to the wave length, L (D/L < 0.15) in the present investigation, (Cheong et al., 1987) the variation of Crrms with kD is not noticeable, indicating that kD does not have any effect on CTrms. This type of trend for CT~ms with kD is seen for all the e/D values encountered in this study. Comparison of regular and random wave force tests The time histories of water particle kinematics in the random wave force tests were generated using the numerical transforms proposed by Reid (1957), based on linear random wave model. The Cr~m~ for submarine pipelines in random waves is also defined RMS Coefficients for submarine pipelines 75 75 60 -@ 45 i Lr~ 30 LD i ~ 65 60 d/D = 90 d / D = 8.0 45 <_) 3O 15 15 r 0 0 5 I0 ~ 0 15 I " l ~ 5 20 K 75 (a) e/D = 0 . 2 5 6o4I%s L) 15 i0 d / D = 7.0 :=1440 - 3980 Legend 30 0 10 5 15 20 • CFHrm s -% CTrms + CLrms 25 75 3°I 25 • d/D k.) 30 ~i0 ~ d / D = 8.0 ~D I,. 0 15 e0 = 90 5 10 20 15 d/D : 0 5 10 15 20 (b)e/D=1.0 7.0 Legend i~ = 1 4 4 0 5 - 3980 " cFn~, -% CTrms + CLrms 0 0 5 10 15 20 25 30 K FIG. 6. Comparison of Cr,mswith CrH,,s and CL,m~ for various d/D. by the Equation 9. The comparison of Cr~m~ for all the d / D investigated and e / D = 0.25 and 1.0 is shown respectively in Figs 9 a and b. The period parameter, K for the random wave record is calculated based on the significant wave concept as, K- UsmaxTp D (12) 66 N. JmHu SHANKAR, HIN-FAI-r CHEONG and K. SUBBIAtl 0 o• 100 e/D = o e5 + ÷ • i o I1"+ • & • 075 + * +• •O~a• ~0 0~ a ~ ~+~ ~ + +. + + + ~ All.++ + • ~,~a.,,o + + + 4+ ^ a lz a+ Legend + + o+ a + + • 4+ * d/D= 9.0 d / D = 7.0 * I I I I 5 i0 15 20 25 K o 150 (b) e/B = 10 & o o~'5 Legend a~ 0 0'~0 e~ ~ +.re :- a ~, +~ _,.~a + ~ + • ÷ ++ ++o~ • • o ooo 0 d/D= N 0 d/D= 80 + d/D=7~0 + k I I t I 5 10 15 2O 25 K FIG. 7. V a r i a t i o n o f ..XC..... with K. in which U........ is the amplitude of the inline water particle velocity for a significant wave having the significant wave height, H,. and period, Tp corresponding to peak frequency, fo. The wave height Hs is defined in terms of the zeroth spectral moment, m • as Hs = 4 \ m , , • (13) The comparison between the regular and random wave force tests results is good inspite of the slight overprediction of C r , ms in the random wave force tests. RMS Coefficients for submarine pipelines 67 75 Legend A (a) e/D = 0 2 5 + kD = 0 d/D = 9 0 ~* o 30 kD = 0.20 kD== 0 156 [-~ L) k D = 0 126 10 x ° Xoo kD=Oll kD = 00B8 o I I I I 25 5 7.5 I0 o o I 125 k D = 0 O? t5 "/5 Legend (b) e/D -- 0 a5 o~ ~ [... L) d/D OO + ¢ O A' A+ ,CO+ O • 8.0 = k D = o 29 kD = 0 gO5 kD = 0161 kA X O ¢ x = 0 131 kD = 0 I07 O O kD = 0 094 1 0 ' I i 2.5 5 7.5 I I 10 i 125 o i 17.5 15 kD = 0 073 20 75 Legend (c) e/D = 0 2 5 o d/D "~o. L~ +x ~ + 0 I I 5 10 = • 7.0 o o x kD= 025 kD = 0 151 • kD = 0 125 x kD = 0.11 o kD = 0 078 kD = 0 098 o I I 15 20 25 K FiG. 8. Variation of Cr..~ with K and kD. 5. CONCLUSIONS This study investigated the regular and random wave induced forces on submarine pipelines fixed near a plane boundary. The total rms coefficient, Crrms is correlated with K, e/D and diD. The results of the present study can be summarised in the following: C r . . s decreases with K, for lower values of K and exhibits a local increase in the 68 N. Jo[}tl SHANKAR, HIN-FAn CHEON(i and K. Su~mAa 75 7+t 60 • 60 d/D : 90 ["3O d/D 45 30 t) : 8 o I 15 0 +, 0 0 5 10 15 lO 5 20 15 20 K 75 60 (a) e / D = 025 d/D : 70 45 Legend 3O + 15 l b ~ 0 5 Re,~ular w a v e tesLs{ff =1440 - 3 9 8 0 I R a n d o m wave tests(@ =1400} amSlm,~a-+_,.,.41Jl, A-.%I~, 10 15 20 25 30 75 25 60 k, ["30 t.) +° f %.~ t5Ik~ z0 d / D = 9.0 GO 5 5 0 5 10 15 o 20 5 to 15 20 25 K 15 10 : 80 10 15 ~ d/D • (b) e / D = 1 0 d/D = 70 b-, Legend 5 • 5 10 15 q~l'AI 20 II It 25 R e g u h x r w~ve te~t.~{~ =1440 - 3980} .a Randon~ w~ve te~,ts{~ =1400) 30 K Fro. 9. C o m p a r i s o n of C~r,,, o b t a i n e d from r e g u l a r a n d r a n d o m wave tests. value between K around 5 and 15 and then decreases gradually with further increase in K. Crrms decreases with decrease in d/D for any given values of e/D and K. CTrmsdecreases with increase in e/D for any given values of d/D and K and exhibits a local increase in the value between K around 5 and 15 depending on e/D. For lower values of e/D, the local increase of Crrms decreases with increase in e/D and is shifted towards higher K values. For larger values of e/D, this type of trend for CTrms is not clearly noticeable. RMS Coefficients for submarine pipelines 69 T h e m e a s u r e d Crrms is in g o o d a g r e e m e n t w i t h t h e c o m p u t e d Crrm~. T h e i n l i n e r m s c o e f f i c i e n t , C r , r,,~ is v e r y m u c h c l o s e to Crrm~ w i t h t h e a v e r a g e a b s o l u t e r e l a t i v e d i f f e r e n c e o f a b o u t 7 . 5 % at l o w v a l u e s o f K. 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