Cabling in Multiple Fall Ropes of Lifting Systems for Deep Water

Transcription

1 OIPEEC Conference Stuttgart March 015 C. R. Chaplin The University of Reading / Reading Rope Consultants LLP, UK Cabling in Multiple Fall Ropes of Lifting Systems for Deep Water Summary Cabling is the metastable twisting together of multiple falls of rope caused by torque in the ropes induced by the applied tension, but resisted by the inclination of the ropes to the vertical. Conventional analysis of cabling assumes that rope tension is uniform, ropes are straight from the point of suspension to a sheave at the hook, and rope separation is the same at the top and the bottom. When, as in deep water operations, the weight of the rope results in a tension gradient from top to bottom, not only must a different approach be adopted in calculating rope torque, but the rope no longer falls in a straight line: it is a catenary. A further complication arises when, in an attempt to improve the resistance to cabling, the rope spacing is increased at the top. This paper addresses all these issues, explaining how to analyse this important problem. 1 Introduction As operations for the exploration and recovery of oil and gas from beneath the ocean extend to ever greater depths problems associated with lowering and raising heavy loads to and from the seabed become ever more challenging. Larger diameter ropes are called for, with multiple falls and even multiple ropes. Loads up to 1000 tonne, or even more, must be handled in depths as great as 3000 m. Rotation of the suspended load induced by torque generated in the rope is a major concern, and a potential problem even with the best designs of low rotation constructions. Perhaps the greatest concern is the possibility of cabling in a system of multiple falls. Cabling (as illustrated in Figure 1) occurs when the stable configuration of multiple falls rotates beyond 90 at which point the resistance generated by the geometry of the inclined falls reaches a maximum, and at mid depth the ropes move closer together until at a rotation of 180 they touch and resistance to further rotation falls to a low value. Rotation then continues until the driving torque from the rope has fallen to a level that it balances the resisting torque generated by the couple of the product of the horizontal component of tension, and the centre-to-centre separation of the ropes where they touch. Cabling analysis is presented below, but simple actions to prevent its occurrence include reduction of the net torque Figure 1: Cabling. Figure 1: Cabling. 47

2 Cabling in multiple fall ropes of lifting systems for deep water generated in the ropes and increasing the separation between the falls. Reduction of rope torque can involve choice of constructions which generate lower torque or use of pairs of ropes which have matched left and right hand lays. Increased separation can be achieved by the use of spreader beams, with sheaves at both extremities, or even devices to pull one fall sideways near its upper connection. Onshore, with tall cranes, a tag line can be used to control load rotation, but the use of any external control is both difficult and costly in deep water, especially since it is likely to require a second support vessel. The recovery from a cabled situation is not straightforward and depends on the configuration of the system. Raising the suspended load will increase the angle of rope entering the cabled section which increases the horizontal component and thus resisting torque. The rotation progressively reduces but the final stages of untwisting are potentially dynamic as the maximum resistance is again recovered and rotation snaps back to a lower value. In very deep operations with many turns of cabling, and where there is a single rope in a two fall configuration with return sheave at the bottom, to raise the end load the rope in the cabled section must slide past itself, this involves friction and there is the possibility of the cable actually inducing slack in the fixed-end fall of the rope. The particular subject matter of this paper is the analysis of cabling in very deep water where rope weight introduces significant differences in tension between the top and the bottom, when the normal simplifying assumptions no longer hold. But first the simple analysis of cabling will be reviewed. Cabling analysis.1 Conventional simple analysis of cabling Previous analysis of simple cabling was presented by British Ropes [1], and later by Hobbs et al. []. Post cabling behaviour was considered by Bradon et al. [3]. The simple cabling equations are based upon assumptions that rope tension is constant throughout each fall, the rope separation at the top is the same as at the bottom (though cabling with different separations has been analysed in []), and the fall is large compared to the separation between the ropes when parallel. In this review only a two fall system is considered with a sheave of diameter D at the lowest point in the system, at a depth of L below the upper rope attachments which have the same horizontal separation, D. Prior to any rotation of the sheave about a vertical axis, the two falls of rope are vertical at a tension, T, assumed to be uniform throughout the two falls. Figure is a schematic representation of the sheave at the lower point in plan view. In response to torque, M, applied to the system (usually the sum of the torques generated in the ropes in response to the applied tension) the sheave rotates by an angle θ. The analysis then determines the resisting torque associated with this rotation which is provided by the moments of the horizontal components of rope tension about the vertical axis through the sheave centre. From the geometry in Figure the moment arm for each rope can be derived. 48

3 OIPEEC Conference Stuttgart March 015 C B D/ θ A D/ O E Figure : Schematic plan view of the sheave both unrotated (AOE) and after rotation through an angle of θ (COF). OB is perpendicular to AC and the PCD of the sheave is D. F Figure 3 is a schematic representation of a side elevation of one rope fall looking in direction BO in Figure. This figure informs the derivation of the horizontal component of rope tension, in terms of the sheave rotation and suspended length. In practice the sheave will rise slightly as a result of the inclination of the rope to the vertical. No attempt has been made to reflect this small vertical movement in either of Figures 1 or since it has no bearing on the static equilibrium analysis employed here this movement is significant in an energy-based analysis (as in []) and much larger for suspended rope lengths typical of normal crane operations than for deep water operations. H L C A Figure 3: Side elevation of suspended rope, in direction BO from figure. From Figure the moment arm for each rope is given by: D BO.cos / (1) 49

4 Cabling in multiple fall ropes of lifting systems for deep water and the horizontal offset AC is given by D AC.sin / D. sin / () From Figure 3 the horizontal component of the rope tension, T h, is given by: T h AC T. L D T..sin( ) L (3) The resisting torque is then twice the product of the horizontal component of tension and the moment arm, and given by: T. D resistingtorque..sin /.cos / L T. D.sin (4) L Assuming that the torque driving the cabling is derived solely from the rope torque this is given by: cablingtorque. T. d. c (5) where d is the rope diameter and c 1 the torque factor, as defined by Feyrer and Schiffner [4]. Note that the resisting torque obtains a maximum at a rotation of 90. Providing the cabling torque is less than this maximum resistance, equilibrium can be established at a value of θ between 0 and 90 given by: sin 4. L. d. c 1 1 (6) D When the cabling torque exceeds the maximum resisting torque at 90, rotation becomes unstable and will increase to form the cabled structure as illustrated in Figure 1. During this increase in rotation the resisting torque falls sinusoidally according to equation 4, while the net cabling torque falls with increasing rotation at a rate determined by the torsional stiffness of the rope. Once the two falls touch at mid depth the resisting torque consists of a moment arm equal to rope diameter multiplied by a horizontal component of rope tension which progressively increases as the length of the cabled section increases and the inclination of each part of the rope to the vertical increases. The details of this mechanism are described by Bradon et al.[3] who also show experimental measurements which agree well with theoretical predictions, as reproduced here in Figure 4. 50

5 Torque (Nm) OIPEEC Conference Stuttgart March Rotation (deg) Measured Pre-cabling theory Post-cabling theory Figure 4: Example comparison of theoretical prediction and measured torque as a function of rotation. The rotation was controlled to prevent unstable rotation at 90, and torque measurement set to zero at 0 thereby cancelling out initial rope torque. Reproduced from [3]. Reference [] also shows how the simple analysis can be extended to systems with unequal top and bottom spacing by replacing the D term in equation 4 by the product of the (lower) sheave diameter and the upper separation between the two falls. This result can be derived using a very similar static equilibrium analysis.. Calculation of rope torque in deep water With an increasing length of suspended rope as encountered in offshore operations in deep water, the weight of the rope results in a significant difference in tension between the top and the bottom. This complicates the determination of the rope torque driving the cabling and the simple expression in equation 5 is no longer applicable. Since the torque must be the same throughout the full length of each fall, the rope rotates to establish this equilibrium torque, with the upper part untwisting and the lower part twisting up 1. The linear model developed by Feyrer and Schiffner [4] provides a means of determining the rotation profile with depth and thus the equilibrium torque. The axial torque, M, is given by: 4 M c1. d. T c. d. T. c3. d. G. (7) s s where c 1,, 3 are constants for the rope, d is the rope diameter, T the tension, G is the shear modulus of the wire, and /s is the local twist in radians per unit length. This model seems to work for all wire rope constructions, including low rotation multi-strands provided 1 The same situation occurs in deep mineshafts where the effect has been given much attention, as for example in [5] and most recently in [6]. 51

6 Cabling in multiple fall ropes of lifting systems for deep water the tension is high enough, and twist low enough, to ensure the rope is in the linear regime (i.e. no slackening of the outer layer). For a uniform axial torque, M, and any given local value of the rope tension, T, local twist can be determined. Thus: M c1. d. T 4 s c. d. T c. d. G 3 (8) The boundary conditions for rotation (the integration of twist) over the rope length are that there is no net rotation difference between the top and the bottom, from which the required axial moment can be established. However, integration of this expression over the suspended rope length has eluded the author so an incremental spread sheet approach has been adopted whereby working from a zero rotation at the lower end, for sequential finite increments in rope length, mid-increment tension and twist can be calculated based upon an initial assumed value of rope torque. This twist is assumed constant over the length of each increment, and rotation at the top of the increment with respect to the bottom is thus calculated. This iteration is repeated until the top of the rope is reached at the sea surface. A goal seeking method (or a manual iteration) can then be performed to converge on the value of rope torque which gives the correct boundary condition at the top. In practice, it has been found from consideration of a number of specific cases (an example of which is given below in section 3) that a very close approximation to the torque calculated by this method is obtained by simple application of equation 5, using the rope tension at the mid point of the rope. This does not give an indication of the level of central rotation however, for which equation 8 and a full twist analysis must be used..3 Calculation of resisting torque of ropes in a catenary When a load attached to the lower end of a (heavy) rope hanging from a fixed point is deflected laterally, the mathematical form of the rope shape is a catenary. The relevant standard equations which describe this situation both geometrically and as regards its mechanics can be found (for example) in [8] or [9]. The equations relevant to this particular problem are given below: x y a. cosh x a (9) s a. sinh a (10) T w. y (11) T v = w.s (1) T h = w.a (13) Feyrer [7] has recently published links to spreadsheet methods for determination of torque and twist in suspended ropes where rope weight is significant. 5

7 x = c x = c + horizontal offset OIPEEC Conference Stuttgart March 015 where x and y are the Cartesian coordinates (with y positive upwards); a is the catenary parameter (and the value of y at the lowest point, where x = 0); s is the distance along the catenary from the lowest point; w is the weight per unit length of the rope; T is the local rope tension and T v and T h are local vertical and horizontal components respectively. Note that the horizontal component of tension, T h, is constant and the axial rope tension increases by amounts equal to the weight of the equivalent vertical length of rope. Further note that (obviously) T v is zero where x = 0. Here because there is an attached load at the lowest point of the rope, we are only concerned with a part of the full catenary. Figure 5 shows the relevant geometry and parameters. y = a + b + water depth y At the top, T v = rope weight + half attached weight T h y = a.cosh(x/a) y = a + b sheave T h T v y = a Figure 5: Geometry and definitions of the rope catenary. The solid red line represents the rope, and the dashed red line the segment of the catenary of which the rope forms a part from its origin to the lower end of the rope at x = c. x The lowest point on the rope is at x = c, y = a + b, where in this instance it is tangential to the sheave on the lower block. The upper end of the rope is at x = c + e (the horizontal offset of the rope), y = a + b + water depth. At the sheave, the vertical component of rope tension, T v, is equal to half the total attached load. This figure and the associated catenary equations are relevant at each stage of cabling rotation (each forming a catenary having its own parameter). Where the rope falls are vertical prior to any rotation, the horizontal offset is at this stage zero and there is effectively no catenary, but as an 53

8 Cabling in multiple fall ropes of lifting systems for deep water increasing rotation of the sheave is considered and the offset increases accordingly the catenary form becomes relevant in determining the horizontal component of rope tension. For cases in which the separation of the falls is greater at the top than at the bottom, there is an initial horizontal offset, and the catenary form is relevant from the start of rotation. It may also be noted that in all deep water situations where rope weight is a significant part of total tension, the horizontal offsets, whether due to differences in separation or sheave rotation, are very small in relation to suspended rope length. Whilst it is therefore tempting to dismiss the catenary form, it is the purpose of this paper to explore the magnitude of the effect and the extent to which slight differences in rope inclination at the sheave affect the magnitude of restraining couple. The quantity required to determine the restraining couple is the horizontal component of rope tension, T h, which, from equation 13, is the product of rope weight per unit length and the catenary parameter a. At any level of sheave rotation there is a unique solution for this parameter determined from the horizontal offset (e), water depth (wd) and rope weight, but a closed form (algebraic) solution of the hyperbolic functions is not possible, and so the parameter must also be obtained by goal seeking or iteration. An initial estimate is needed for a (a est ). A suitable value can just be guessed, or calculated by considering the rope as a straight member and taking moments about the upper end to first obtain an estimate for T h, and then use equation 13 to obtain a est. This gives: endload e wd. T h.est. e wd. w. then, a est T h.est w (14) Using this estimated value for the catenary parameter in combination with the known value for half the attached end load, c the value of x at the lower end of the rope where it contacts the sheave, can be estimated using equations 10 and 1. So: c est a est.sinh 1 endload. w. aest (15) Using equation 9 an estimate for the water depth, wd, is given: wd est a est cest.cosh a e a est est c. cosh a est est (16) A goal seeking (or simple iterative) procedure can then be used to adjust the estimated value of a used in equation 15 and then 16, until the estimated water depth converges to 54

9 OIPEEC Conference Stuttgart March 015 the required water depth. It is then a simple matter to determine the correct horizontal component of rope tension, (T h ), using equation 13. The next stage is the calculation of the moment arm of the rope acting on the rotated lower block/sheave assembly. In situations where the rope spacing at the top and the bottom are the same, the moment arm of the resisting couple will be the same as in equation 1 for the simple case, and the resisting couple is given by: D resistingcouple. Th..cos T. D.cos h (17) However where the upper spacing has been increased, to assist in preventing cabling, the more complex geometry must be considered, which is represented by the schematic diagram in Figure 6. The schematic plan is to be interpreted in terms of rotation of the vertical plane of the rope catenary which is pulled sideways at the bottom, with rotation of the sheave about a vertical axis. Point H is vertically below the highest point of the rope; the unrotated sheave is represented by AOE, and two rotated positions are represented by BOF and GOJ (the latter having the same moment arm, CO, as the former but with the sheave in position GOJ). The sheave diameter is D so AO = BO = GO = D/. C G β B β α θ H e A D/ O E F Figure 6: Schematic plan view of lower sheave rotation with greater rope separation at the top. J HA on the diagram is the initial horizontal offset of the catenary (e 0 ) given by: 55

10 Cabling in multiple fall ropes of lifting systems for deep water e 0 upperrope spacing D (18) For rotation of the sheave by θ to BOF, the moment arm OC is given by: D OC e 0.sin (19) By the sine rule for triangles, e 0 D / sin D / HB sin sin 1 D so sin e0.sin / D / and HB, the horizontal offset at sheave rotation of θ D sin. sin or similarly D D HB e0.cos. cos (0a) D D HG e0.cos. cos (0b) These equations were found most easily quantified by tabulating solutions for different values of α. To quantify the resisting couple, equations 0a or 0b are needed to determine the catenary parameters using equations 9 to 16 and thus the horizontal component of rope tension, which is then multiplied by the moment arm from equation For the limiting situation where the sheave rotation, θ, is 90 the calculation is simplified and the associated horizontal offset is given by: and the moment arm is: D D e 90 e0 (1) 3 Note that this complicated trigonometrical analysis is not required for the simple analysis with straight falls, as in reference [], since the unknowns which must be quantified here cancel out in the simple analysis, thus avoiding any need for their determination. 56

11 OIPEEC Conference Stuttgart March 015 D e0 D. e 90 () 3 Results The numerical results presented below all relate to a hypothetical (but realistic) case with the following parameters: rope diameter 76 mm linear rope mass (submerged) kg/m water depth 1000 m min. attached mass (block + sheave) 10 tonne max. payload 100 tonne sheave PCD 1.9 m (D/d of 5:1) upper rope spacing either 1.9 m, or, 6.0 m rope constructions considered A: 6x41 ordinary lay with IWRC B: low rotation 35 strand rope rope 4/16 from [10] torsion constants (as in equation 7) c 1, c and c 3 for rope A 0.077, 0.16, (measured for a 77mm rope) for rope B 0.05, 0.11, (for 16 mm rope from [9]) 3.1 Rope torque driving the cabling process For both minimum attached loads and maximum attached loads, rope torque has been calculated based upon mean tension alone (equation 5 but for one rope) ignoring any rotation within the suspended length. This may be compared with torque using the incremental method described in section. above from which the maximum rotation can also be determined. For the limiting value of cabling rotation at the lower sheave (-90 : i.e. untwisting), central rotation has been calculated for this revised boundary condition and torque has been recalculated to indicate the effect on the torque which is driving cabling as the lower block rotates to the point of instability. The results are set out in Table 1. Note that while torques generated in the six strand rope are around three times those of the multi-strand low rotation rope, the ratios of mid-depth rotations are much more marked being of the order of a factor of 50. This is due to the large difference in torsional stiffness (c 3 ). For the same reason there is a significant fall in torque in rope B associated with the 90 rotation to the point of cabling instability. By comparison, with such a high level of rope twist, the change in torque associated with the same rotation in rope A is imperceptible. 57

12 Cabling in multiple fall ropes of lifting systems for deep water Rope A Rope B min. loading max. loading min. loading max. loading Rope Tension - kn max. (top) min. (bottom) mean (mid depth) Torque - Nm (c 1 mean) Incremental model Torque Nm max. (central) rotation 17.0 turns 13.0 turns torque at 90 block rotation Table 1: Results of calculations of rope torque. In calculating susceptibility to cabling it is safe to assume that rope torque remains constant as the lower block rotates. This involves a very small error for six strand ropes, but a somewhat larger error for low rotation ropes. The simplification is acceptable as it results in over-estimating the rope torque at the critical 90. Furthermore whilst most rope manufacturers are willing to provide values for torque factors (c 1 ) the torsional stiffness terms required for this calculation are generally unknown territory Resisting couples Extensive numerical examples have been analysed to evaluate the resisting couple for different conditions and as a function of lower sheave rotation. Somewhat surprisingly it has been found that in all examples, even with the complexities of catenary rope falls and greater separation of the falls at the surface, this relationship always follows a sinusoidal profile as explicitly derived in the closed form solution for the simple case. As a result, in practice, it is only necessary to calculate the couple for the limiting condition of 90 rotation given by the product of the moment arm, from equation, and T h from catenary analysis using the horizontal offset from equation 1. Then the sinusoidal form can be applied for lower values of rotation. This considerably simplifies the calculation routine and subsequent assessment of vulnerability to cabling. 4 In passing it is worth noting that the author has had considerable difficulty obtaining from manufacturers any validated torsion constants for their ropes other than nominal values for the torque factor, frequently accompanied by assertions that with use the factor for the rope as installed will fall. In addition to a torque factor some manufacturers do provide data for some constructions on rotation of an unrestrained end load. But torsional stiffness remains numerically elusive. 58

13 OIPEEC Conference Stuttgart March 015 The same example as in section 3.1 will be considered, focusing on the resisting torque at 90 lower block rotation. Parallel falls as well as falls with a higher surface separation will be considered. In section 3.3 the results will be compared with the cabling torque generated by ropes with different torsional properties. For comparison, as well as calculating the maximum resisting couple based upon catenaries, alternative simple procedures have been used. For the case of two falls with the same rope separation at the top and the bottom, results are presented using equation 4, but with mean tension and minimum tension as alternatives. In the example with increased rope separation at the top, as well as calculating using the simple method with the two different tensions, alternative approximation methods are presented for the D term using either the product of the top and bottom spacing (the square of the geometric mean), or, the square of the average (square of the arithmetic mean). Of these two alternatives the former, based on geometric mean, was recommended in reference []. The numerical results are presented in Table. Equal 1.9 m separation, top and bottom Maximum Resisting Couple Nm Minimum Loading 10 tonne attached Maximum Loading 110 tonne attached simple method using bottom tension simple method using mean tension full catenary method Separations: 6 m top, 1.9 m bottom product (D =6 1.9) mean (D =((6+1.9)/) ) product (D =6 1.9) mean (D =((6+1.9)/) ) simple method using bottom tension simple method using mean tension full catenary method Table : Results of calculations of maximum resisting couples (in Nm) at peak (90 ) rotations using different procedures, and for the case defined in the text. Of the different results presented in Table those based upon the full catenary must be considered the most accurate predictions, since they most accurately reflect the real situation. Of course in practice other factors can be present which might invalidate or at least moderate some of the assumptions implicit in the calculation method. 59

14 Cabling in multiple fall ropes of lifting systems for deep water For the two equal separation cases, torques calculated with the simple method and using the bottom rope tension (which seems most appropriate since it acts at the sheave) significantly underestimate the resisting couples. This illustrates the whole basis of the argument for using the catenary method which correctly models the shallower angle of the rope at the sheave. Using the mid-depth tension in the simple method gives closer agreement, though over-estimating the resisting couples and thus increasing the risk of cabling: but there is no logical justification for using this value of tension in the calculation. In the case of unequal spacing, the observations are similar whether replacing D in equation 4 by the product of the upper and lower spacing (the correct solution for the simple case), or the square of the mean spacing. But as in the equal spacing case, calculations based on minimum tension are too low, while those based on mean (middepth) tension are more accurate but too high. And furthermore there is no justification for using mean rope tension for calculating the couples. It is also clear from a comparison of values in Tables 1 and that for the operational conditions considered in the example, not only is the six strand rope unsuitable but even the low rotation rope requires the benefit of increased rope separation at the surface to provide the operator with a reasonable degree of confidence of avoidance of cabling. It must be emphasised that all these calculations are based upon a single set of parameters as regards depth, loading, sheave diameter and upper rope spacing, and for this reason cannot be assumed to be other than broadly indicative of the kinds of inaccuracies likely to result from using approximations based upon any simple method of cabling analysis. 3.3 Combining rope torque and resisting couples The susceptibility of a system operating at specific depth with specific loads can be assessed by comparing the predictions for the rope torques driving the rotation and the geometrically derived couples resisting rotation. This is best achieved by plotting the two parameters as a function of cabling rotation; and such a plot can easily illustrate the impact of using ropes with different torsional characteristics. Figures 7 and 8 show results for the system defined above at minimum and maximum loading respectively. In Figure 7 the results are presented for the case defined above with minimum loading, 6 m rope separation at the surface, resisting couple calculated using the full catenary calculation, and rope torque levels indicated for three different values of torque factor (c 1 ). The reductions in rope torque with twist presented in Table 1 are have been included for information since these low torque factors are only obtainable with multi-strand constructions having the relevant high torsional stiffnesses. Equilibrium rotations are indicated by the intersection of the relevant rope torque with the cabling resistance curve. Note that a rope with an assured torque factor of less than % is required for a reasonable margin of safety under the specific conditions considered. If a safety factor of (applied to the maximum restraining torque) is considered, then in terms of rotation, given the sinusoidal relationship, this is equivalent to designing for a maximum rotation of

15 Resisting torque & rope torque - Nm Resisting torque & rope torque - Nm OIPEEC Conference Stuttgart March torque factor : 3% torque factor : % torque factor : 1% sheave rotation - degrees Figure 7: Sinusoidal cabling resisting couple (torque) and rope torques for different torque factors at minimum loading, for the case defined in the text. The rope torques are shown as decreasing with rotation as indicated in Table 1, for low rotation ropes. The resisting couple is for the system with increased (6 m) rope spacing at the surface torque factor : 3% torque factor : % 1000 torque factor : 1% sheave rotation - degrees Figure 8: Sinusoidal cabling resisting couple (torque) and rope torques for different torque factors at maximum loading, for the case defined in the text. The rope torques are shown as decreasing with rotation as indicated in Table 1, for low rotation ropes. The resisting torque is for the system with increased (6 m) rope spacing at the surface. Observations regarding the results presented in Figure 8 are similar to those for Figure 7 except that with the 100 tonne payload attached the risk of cabling is slightly reduced as indicated by the intersections of rope torque with resisting couple being at lower rotations. 61

16 Cabling in multiple fall ropes of lifting systems for deep water 4 Conclusions 1. The simple analytical approach to rope cabling has been reviewed, with results repeated from [3] which provide experimental support for the theoretical predictions of resisting torques generated both in the initial stage (from 0 to 90 ), in the immediate instability phase (from 90 to 180 ) and in the subsequent fully cabled stage.. The simple method of analysis is not suitable for assessing the risk of cabling in deep water applications where tension changes significantly with depth due to rope weight. As shown in the examples presented, the resulting errors are greater when the end load is small in relation to rope weight as the catenary will then be more evident at the bottom of the rope. 3. Methods for quantifying rope torques have been explored in situations where the differences resulting from the weight of the suspended rope length are significant. An incremental model which allows for mid-depth rotation of the rope to achieve uniform torque, indicates that an approximation based upon tension at mid depth and the torque factor (c 1 ) is sufficiently accurate. However the more representative full analysis quantifies rope rotation profile with depth and also allows calculation of rope torque reduction as the attached assembly rotates by changing the boundary conditions for rope rotation. However an issue here is that the necessary full torsional characterisation of the rope is seldom available when designing a system and choosing rope constructions and diameters. 4. The effect of twisted rope being transferred round the sheave from one fall to the other during either lowering or lifting has not been considered. It is thought a reasonable assumption that the associated effects will be reversible, and balance out between the two falls. However the potential handling issues resulting from twisted rope being stored on the drum of the hoisting system are another matter, and details of a sequence of operations if at different depths might become relevant. 5. In considering the relaxation of rope torque during the rotation towards the limiting 90 it was shown that when using low rotation ropes with low torque factors and high torsional stiffness the effect is sufficiently large for it to be worth quantifying. But for much more flexible six strand ropes, with their higher torque factors the effect is negligible. Ignoring any such reductions, whatever the construction, constitutes a safe assumption. 6. Calculations made by the author, but not reported in detail, have demonstrated that even with a full catenary analysis and asymmetric rope spacing (e.g. higher at the surface than the bottom, as in the example examined) the increase of the cable resisting couple with rotation remains sinusoidal between 0 and 180, thus reaching the maximum, and limiting point of stability, at 90 rotation. This simplifies any modelling of such systems since the catenary geometry need only be solved for the 90 case, avoiding some of the additional geometric complexities of lesser rotations. The sinusoidal profile can then be assumed for rotations up to 90. 6

17 OIPEEC Conference Stuttgart March It has been described how a full catenary analysis can be made, but at present there is no closed form solution (i.e. a simple equation cannot be presented as is the case for the simple method) and each case must therefore be analysed independently and numerically. This is time consuming but not too difficult once a procedure has been set up. 8. All the numerical values derived here relate to a single example with only maximum and minimum loading conditions, two rope constructions, and two rope separation configurations being considered. In practice a very wide range of operational scenarios is possible, and for such different conditions only broad (and rather obvious) conclusions can be presented: a. depth makes a considerable difference to the risk of cabling; b. rope construction is all-important and the true values for rope torque factors are essential in making a reliable assessment of the risk; c. increasing rope separation is the most effective way to reduce the risk of cabling, and whilst not so effective, increasing spacing at the top only is useful (even if it makes calculation more complex); d. approximate methods for deep water based upon a simple (i.e. straight rope) analysis are not sufficiently accurate to inform design decisions (and the apparently more accurate approach of calculating couples using mean rope tension, as opposed to the minimum, is not justifiable). 9. It is recommended that when designing a deep-water system for lowering and raising heavy loads a reasonable margin is maintained between the equilibrium rotation and the metastable 90. Things happen slowly underwater, but the objects being handled typically have high inertia, so that once a rotation has initiated overshoot must be considered inevitable. This can transform what appears stable on the basis of the equilibrium rotation, into dynamic overshoot beyond the critical 90. Once cabling begins where such great rope lengths are deployed, the postcabling equilibrium rotation is likely to involve many turns of cabling. Such a state may not be easily reversible especially considering the frictional interactions between rope falls in the cabled region. It would seem sensible to require a minimum factor of safety on maximum restraining torque of at least, which would indicate designing to a maximum equilibrium rotation of 30. Where there is uncertainty as to other external forces, which might add to rope torque in driving the cabling (such as current induced forces), a higher factor should be considered. 10. Only cabling of two falls has been considered in the example above, but the method described for catenary analysis can readily be extended to systems with higher numbers of falls. 5 References [1] British Ropes Limited, Ropes for Oil, British Ropes publication No.118,

18 Cabling in multiple fall ropes of lifting systems for deep water [] Hobbs, R.E., Sharp, D.M. and Walton, J.M. Cabling of Crane Ropes, Wire Industry 60, (1993), [3] Bradon, J.E., Chaplin, C.R. and Ermolaeva, N.S. Modelling the Cabling of Rope Systems, J. of Engineering Failure Analysis 14 (007) [4] Feyrer, K. and Schiffner, G. Torque and torsional stiffness of wire rope Part I, Wire, 36(1986) 8, [5] Rebel, G., Chaplin, C.R. and Borello, M. Depth limitations in the use of triangular strand ropes for mine hoisting, Proceedings of Mine Hoisting 000, SAIMM Johannesburg, September 000, [6] Malinovsky, V.A. and Prigoda, A.A. Impact of internal constructional friction upon the torsional behaviour of mine wire ropes OIPEEC Int. J of Rope Science and Technology, 103 (014) [7] Feyrer, K. Wire rope twist calculation programmes OIPEEC Int. J of Rope Science and Technology, 103 (014) [8] Lamb, H. Statics, Cambridge University Press, 191. [9] Inglis, C.E. Applied Mechanics for Engineers, Cambridge University Press, [10] Reinelt, O., Winter, S. and Wehking, K-H. Determination of the Donandt force of rotation resistant wire ropes under dynamic working conditions OIPEEC conference Simulating Rope Applications, Oxford, 013 (ODN0898), Nomenclature a : the catenary parameter a est : initial estimated value of a at start of goal seeking iteration b : vertical distance from lowest point of catenary to lower end of rope at sheave c : x coordinate on catenary to lower end of rope at sheave c 1,, 3 : rope torsion constants (as defined in [4]) d : rope diameter D : sheave PCD e : horizontal offset of rope at sheave from the end at the surface G : shear modulus of constituent wires in a rope (80,000 N/mm here) L : suspended rope length M : axial rope torque s : distance along rope T : rope tension T h : horizontal component of rope tension in catenary (constant) T v : vertical component of rope tension in catenary (function of x) w : rope weight per unit length x, y : catenary coordinates α and β : angles defining geometry of catenary rotations φ : rope twist about its axis θ : sheave rotation about vertical axis 64