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- 1. DESIGN METHODOLOGY FOR UNDERSEA UMBILICAL CABLES bY R.H.Knapp, Professor and T.T.Le, Research Assistant Department of Mechanical Engineering M.J. Cruickshank, Technical Director Marine Mineral Technology Center University of Hawaii at Manoa Honolulu, Hawaii 96822 ABSTRACT Nomenclature Umbilical cables are widely used in the ocean industry for = layer areaexploration and recovery of petroleum and mineral resources. = core areaDevelopment of these cables requires careful design to satisfy = coefficients of radial displacement functionsimultaneously power, communication and strength functions. The = matrix of penalty parameterspresent paper suggests a design methodology to facilitate the = helical wire coveragemechanical design of umbilical cables. A design nomograph and simple = wire diameterformulas suitable for hand-calculation are presented as preliminary = elastic tensile modulusdesign aids. Computer software to assist in finalizing the design is = core compressive modulusdescribed. = factor of safety = core shear modulus = core effective polar moment of inertia INTRODUCTION = cable cross-sectional stiffness matrix = stiffness Today’s umilical cables are as diverse in design as their many = lengthapplications in the ocean including power cables, control cables, tow = cable bending momentcables, remotely-operated vehicle (ROV) cables, subsea wellhead cables = number of wiresand mining cables. Umbilical cables may consist of numerous = number of harmonic termsdissimilar components to conduct electrical power, optical control = matrix dependent on contact angles, pisignals, and fluid power. Also, armoring wires are needed to protect = local displacement vectorthese conductor components. This paper is concerned with the = global displacement vectormechanical design of a composite umbilical cable under the = radius in polar coordinatessimultaneous actions of tensile, twisting, bending, pressure, and = boundary constraint matrixthermal loads. A design nomograph and simple formulas are presented = wire radiusfor preliminary design. Finally, computer models for more refined = pitch radiusanalysis are described. A numerical example is presented to illustrate = global load vectorand verify the accuracy of the design method. = core outer radius = yield strength = penalty function matrixMechanical Failure Modes = cable tension = radial displacement CQmposite cables are subject to electrical, optical and = circumferential displacementmechanical failure modes. This paper is concerned with the mechanicalfailure modes described below. Tensile failure occurs if the tension = helical wire lay angle: t) right-hand lay; (-) left-hand layapplied to the cable produces yielding or fracture of any of its = angle to contact point {Figure 3) = cable straincomponents. Excessive rotation of a free-end cable can produce = 0 : rigid core assumptionexcessive cable elongation and unequal load sharing among the helical = 1 : incompressible core assumptionarmor layers as load transfers from layers that are loosening to those = circumferential angle in polar coordinateslayers that are tightening. Excessive torque of a fixed-end cable can = total potential energylead to cable hockling (unstable looping) if the applied tension is = axial stressinsufficient to maintain a straight cable. Bending failure frequently = cable torqueoccurs near supports, terminations and where the cable bend radius is = cable rotationsmall enough to produce significant bending stresses. Fatigue failure = cable curvature angleoccurs due to the cyclical application of any load, but is most Subscripts:frequently the result of repeated bending, especially near terminationsor supports. Temperature changes can produce damaging internal a = antisymmetric 1 = layer numberstresses, especially for plastic materials and delicate optical fibers. d = dependent layer L = total number of layers e = condensed ring element m = macro-element k = Fourier series harmonic s = symmetric -&7803-M02-8/91/OOm-13 .COO199 EEE 19$1 1
- 2. DESIGN PROCESS The process of designing an umbilical cable is illustrated by theflowchart in Figure 1 and further detailed in Tables 1 to 7.Operational Parameters START To begin the design process, each operational parameteridentified in Table 1 must be defined or estimated. All core Specify Operational Parameterscomponents including the size and quantity of conductors (electrical, (see Table 1)optical and fluid) are selected to meet the needed transmissionrequirements. The desired geometry of the cable suspension, the cableweight and current velocity profiles are to be estimated. To facilitatecable storage, deployment and retrieval from a support vessel, physical Icharacteristics of the cable such as weight, overall diameter, axialstiffness, torsional stiffness, and bending stiffness must be carefullyconsidered. The stiffnesses need to be matched to the cable handlingsystem. Helical strength members having small lay angles providegreater strength and axial rigidity but reduced torsional and bendingflexibility. The opposite is true for large helical lay angles. The designmethod must permit tradeoff studies to satisfy strength anddeformation criteria and handling requirements. Umbilical cables must be designed for repeated deployment andretrieval. Suitable materials must be selected to minimize corrosion,fatigue and surface abrasion. Table 1. Operational Parameters Cable Transmission: Conductors insulation Layers Fluid Conduits Cable Suspension: Cable End Coordinates Cable End Excursions Cable Weight/Length Current Velocity Profile Cable Handling: Cable Weight/Length Overall Diameter Axial, Torsional and Bending Stiffnesses Cable Strength: End Conditions (fixed or free to rotate) Loads (tension, twist, bend diameter, bend cycles, pressure, Prototype Testing temperature) (see Table 7) Factors of Safety FINISH Figure 1. Mechanical Design Flowchart Table 2. Preliminary Design Specify Conductors: Electrical Optical Fluid Armor Area Helical Wire Coverage Torque and Stress Balance Linear Cable Equilibrium Equations Helical Wire Axial Stress Cable Hockling 1320
- 3. Table 3 Cable Cross-Section . Armor Area. The strength members serve to carry the applied cable loads and to provide mechanical protection to the electrical, optical and fluid conductors located within the cable core. For the Geometrical Layout: latter purpose, such strength members are referred to as armor. To Size & Number o Conductors (electrical, optical, fluid) f provide bending flexibility, the armor wires are nearly always helically Insulation Layers (sheaths, jackets) Filler Wires served around the cable core. Armor Wires The nomograph presented in Figure 2 permits rapid estimates Select Materials: of the armor area in any layer. This nomograph is based on a linear Elastic Modulus formulation subject to the assumptions: Yield and Ultimate Strengths Poisson’s Ratio (1) the ends of the cable are fixed against rotation; (2) the core is radially rigid and its axial rigidity is neglected; Nonlinear Stress-Strain Curve (3) each layer of armor carries an equal fraction of the applied tension; Hardness (4) wire bending and torsional stresses are neglected. Thermal Coefficient o Expansion f Specific Gravity To use the nomograph, follow the steps outlined in Table 8. If a standard circular wire diameter, d,, is to be used for layer ’l’, the number of wires, nl, in a layer is computed by the formula, Table 4. Cable Properties Wire Coverage Cable Outer Diameter Cable Weight/Length Structural Stiffnesses (axial, torsional, bending) Helical Wire Coverage. The wire coverage is a measure of how completely the number of wires, nI, fills a layer. For a compact cable cross-section, it is desirable to choose a coverage above 90%, but less than 100%. Wire coverage in this range helps to maintain a stable construction during handling. Nowak[ 151 presented a formula for coverage based on an elliptical approximation to the shape of a helical Table 5. Cable Suspension Analysis wire cut transverse to the cable axis. This coverage formula is sufficiently accurate for cables of practical interest. After some Maximum Axial Tension, rearrangement, Nowak’s coverage formula becomes, Maximum Rotation Maximum Bending Moment Cl[%] = -XI00 n14 (2) Minimum Bend Radius 21rR,coscul Hockling (unstable looping) where RI is the pitch radius measured from the cable axis to the wire axis. Table 6. Computer-Aided Design Torque and Stress Balance. The helically-served components Cable Strain that render the cable flexible also induce a torque as each helical wire Cable Diameter Reduction layer tries to unwind with the application of tension. The net torque Cable Torque or Rotation produced causes a free-end cable to rotate with some layers tightening Component Stresses and Strains and some loosening. This means that some layers will be stressed at Minimum Bend Diameter higher levels than others, thereby lowering the cable breaking strength. Radial Pressure Distribution Furthermore, long cables that are restrained from rotating can develop sufficient torque to produce hockling (unstable loop formation) if the cable tension is relaxed. To minimize the cable torque and potential hockling, contrahelical layers of armor (alternating left- and right- hand lay directions) can be employed in the cable design. Table 7. Prototype Testing For structural efficiency, it is important that each armor layer Conductor Performance: carries a fraction of the total tension proportional to its yield strength. Electrical, Optical, Fluid This condition is termed stress balance. Mechanical Performance: Knapp[7] has demonstrated that the torque and stress balance Deformation, Strength, Fatigue conditions are approximately met if the following equation is satisfied: sinad = - (3)Preliminary Design In this section, a design nomograph and simple linear cable where ad = lay angle of dependent layer, d (usually the outermostmodels, suitable for hand-calculation, are suggested as preliminary layer).design tools. They are used to estimate the size and quantity ofstrength members needed and the overall response of the cable to Linear Cable Equilibrium Equations. The preliminary selectionapplied tension, twist and bending actions. This initial definition 0.f of armor can be improved by use of equilibrium equations [6][11] thatthe cable cross-section is subsequently subject to refinement using relate the applied tension, T, torque, 9, bending moment, M, to the andcomputer-aided design tools. cable axial strain, ec, twist, Aq5/Lc and bend curvature, A$/LC. 1321
- 4. A B C C c --loo0 --goo --800 -- 700 - I 3 --600 --500 2 2 z -400 --300 > + U Y - - l --250 t; -- --200 5 -- -- 175 U - 150 5 - K 0 __ U --125 2 - - 3 L --loo Y m --75 m 9 - U -- 50 Y U - -2 -- 25 Figure 2. Nomograph for Estimating Area of Armor Layer Table 8. Use of Nomograph where the stiffness terms are (1) Specify the factor of safety, FS, on yielding for the cable kT, = c "L [ qAIEI(l-.!-tan2al)cos5al] R C + ACEc tension, T. 1.1 2 RI (2) Select the armor material and compute the ratio, E/S, for each layer. "L kT, = qAIElRlsinalcos2aI (6) (3) Draw a line intersecting the FS, E/S, values and the A-axis 1=1 for each armor layer. (4) Guess at a helical lay angle, a, for each armor layer. "L k,, = e Rc qAIEIRl(1 ---tan2al)sinalcos2al (7) (5) Draw a straight line from the intersection on the A-axis (step 1=1 2 4 3) to the lay angle axis labeled A (right-side of chart) for each armor layer. Select the maximum value which intersects the B- axis. (6) Draw a straight line from the maximum value on the B-axis (step 5) to the lay angle axis labeled B. Repeat this for every layer and mark the intersections on the C-axis. (7) Draw a straight line from each intersection on the C-axis to the Number of Armor Layers line, nL. labeled C. Where these lines cross the Armor Area line determines the dimensionless value, AE/T, for each layer. Knowing E and T, the armor area, A, can be computed. "L = nlEIIl + EcIc (with wire slip) (10) 1=1 where If wires within a layer are of identical size and symmetricallydisposed around the cable axis, the linear equilibrium equations are 6 = 0 : radially rigid core (11)given by = 1 : incompressible core and other terms in the above equations are defined in the Nomenclature. Note in equation (4) that the bending equation is decoupled from the axial-torsional equations. This is a consequence of the previously stated symmetry assumption that is justified if a sufficiently large number of wires occupy a layer. Also, equations (6) 1322
- 5. and (7) reveal that the stiffness matrix is not symmetrical if 0=!;viz., Cable Cross-SectionkT+ # k,c. This is a result of treating the core as an ideallyincompressible material where the core diameter change depends only Based on the preliminary analysis performed in the previouson the axial strain and not on core twist. The bending stiffnesses given section, or following a computer-aided analysis (described below), theby equations (9) and (10) represent the two ideal cases of beam bending cable cross-section is defined (or redefined).(no wire axial slip) and individual wire bending (with wire axial slip).Since real cables experience some slip dependent on the bend The electrical, optical and fluid conductor components arecurvature, the actual value of kM$ will lie somewhere between these specified and geometrically arranged within the cable core. Fillertwo values as has been experimentally verified by Durelli, et al. [4]. wires may be provided to occupy interstitial voids between adjacentFor an improved analysis of bending behavior, Lanteigne[ 1 I] has core components. The core may be provided with a protective outerproposed a wire slip criterion that can be used to select either equation sheath around which layers of helical armor wires are served (for(9) or (IO) for each wire layer in the cable. He also presents stiffness example, see Figure 4). Also, an optional outer jacket may be providedterms in equation (4) that couple the tension, twisting and bending to protect the outer armor layer from abrasion during handling.actions when symmetry cannot be assumed. For preliminary design,however, equation (4) should be satisfactory. In addition, all material properties listed in Table 3 must be specified. For nonlinear materials, a stress-strain curve is needed for the computer analysis. For components such as the core sheath, a Helical Wire Axial Stress. In general, the helical wire will compressive stress-strain curve is preferable. Polymeric or elastomericexperience axial, bending and torsional shearing stresses. For cylindrical layers may be indented by adjacent circular wire layers.preliminary design estimates, only the axial stress component will be Such wire indentation can greatly affect the cable strain and theconsidered here. A simple expression for the wire axial stress has been induced cable torque. To account for this effect, the computerderived in terms of the cable axial strain, the bending-induced wire software described below requires the hardness of the indented layeraxial strain, cable twist and radial deformations of a rigid @=O) or to be specified.incompressible (€+=I) core: Cable Properties Following the flow chart in Figure 1, the cable properties listed in Table 4 should be checked. Wire coverage can be approximated bywhere, preceding f, the (+) and (-) signs refer to the outside and inside equation (2). It is important to ascertain that the cable outer diameterof the bend, respectively, and is geometrically compatible with all handling equipment such as the overboard sheave. Cable weight may be limited by the load ratings of f = 0 (wire slip) (13) the deck handling equipment, and it needs to be specified for a cable = I (no wire slip) suspension analysis. Also needed for the cable suspension analysis are the axial (AE), torsional (JG) and bending (EI) stiffnesses. In terms of the approximate linear equation (4), AE = kTr, JG = k,, and El = kM+. Cable Hockling. A cable that is not designed for torque-balance may develop a sufficiently large torque relative to the appliedcable tension, that relaxation of the tension (momentary slack cable)can result in hockling (unstable looping) as the cable tries to relieve the Cable Suspension Analysistorsional strain energy. If a loop were to develop, reapplication of thefull tension could damage cable components as the hockle curvature is The analysis of a cable suspension is needed to determine theincreased. maximum values of cable tension, rotation and bending moment that the cable must be designed to withstand. Since the cable structural Several investigators have proposed models to predict the properties must be known for the cable suspension analysis and thetension, T, needed to maintain a straight cable configuration for an maximum loads acting on the cable must be known to determine theinduced torque, +. Greenhills formula [ 5 ] , which was also derived by structural properties needed to prevent any component failures, theLiu[ 131 using Timoshenkos buckling formula for a slender rod, design iteration described in the flow chart of Figure 1 is suggested.represents the tension at the onset of hockling, Modeling of cable suspensions has been undertaken by T = -5 2 numerous investigators. It is beyond the scope of this paper to discuss 4EI numerical solutions. The reader is referred to the pertinent literature including Walton & Polacheck[221 who developed a finite differenceBy assuming that the loop diameter must be a magnitude to minimize solution for nonlinear transient motion of cables, and Obrien[l6] andthe torque, Ross[21] obtained, Peyrot[ IS] who developed special cable elements based on the equations of an elastic catenary when bending stiffness is negligible. Leonard[l2] and McNamara[l4] describe the use of finite element analysis including bending stiffness, and Riggs and Leraand[20] proposed an improved cable element for static analysis that includes an algorithm for adaptivewhich is twice as large as was obtained by Greenhill and Liu. discretizations of the cable when fluid drag forces are present.Liu presented the empirical formula, Oliveira, et al. [I71 describe simple analytical methods involving the cable catenary that can be used to find the maximum tension and T = 332 - (experimental) bending moment of flexible pipes subject to uniform fluid drag. This SEI latter method may be suitable for preliminary design estimates.which is halfway between the values given by equations, (14) and ( 1 9 ,and therefore might be used for preliminary design. Recently, Computer-Aided DesignCoyne[3] p.roposed a more rigorous model that depends on cable length.His result is also bounded by equations (14) and (15). The SAC (Stress Analysis of Cables) computer program has been developed as a design tool for the cable industry [9][10]. After The flexural rigidity, EI, appearing in these equations can be establishing a preliminary design as previously described, this softwareestimated by the stiffness factor,? ,k in equations (9) and (10). can be used to rapidly perform parametric studies of geometrical and material quantities to achieve an optimal design. 1323
- 6. Fundamentally, the SAC model consists of adjacent oroverlapping concentric circular layers of any thickness. A layer can bea thick-wall cylinder or filled with equally spaced circular or keystonewires. The wires can be solid or fibrous sections. The geometry of themodel is defined by layer inner and outer diameters, wire sizes and laylengths. Structural properties of all layers are computed by theprogram; however, the user can override this feature and manuallyinput structural properties. For example, this would be useful to modelmore accurately a copper strand. For helically-served layers, theprogram assists in selecting the correct number of wires to "fill" a layer e=ofor the desired coverage. Moreover, since the coverage formula usedis geometrically exact, it will give the correct coverage for wire layershaving large lay angles. Loads can act alone or in combination with other loadsincluding axial tension, twist, bending, clamping, and internal pressure.Hydrostatic pressure and temperature change can be specifiedindependently for each layer. For bending analyses, the program Ecomputes the minimum bend diameter for layer yielding as well as thebend diameter that produces circumferential wire contact at the insideof the bend where wires are "pinched." Figure 3. Macro-Element for Typical Cable Component The highly nonlinear layer equations are solved so as to satisfyall equilibrium conditions and deformation compatibilities in the axial, whereradial and circumferential directions. Nonlinearities include nonlinearstress-strain curves, large deformations such as rotation and plastic Ig(r) = b, + bzr + b3rzindentation of circular wires into adjacent (inner or outer) cylindricallayers. For wire indentation, any initial radial indentation and the V&r) = b, + bsr + b6r2Durometer hardness of the indented cylindrical layer must be specified. ua(r) = b7 + b8r + b9rz Va(r) = blo + b l l r + bl,r2 The SAC model computes layer and cable weights based on thespecific gravities of wires, wire jackets, sheaths, jackets and voidfillers. Program data files may be converted among the English, and the upper limit, N, is chosen to satisfy convergence of the series.Metric, and SI measurement systems. Plots showing the exact scaled The subscripts s and a represent the symmetric and antisymmetricgeometry of the cable cross-section and analysis results are generated terms relative to the $=0 axis in Figure 3.by the program. Using equations (17) and (18) and following the stiffness formulation for a ring element [2][24], element stiffness matrices representing symmetric and antisymmetric terms can be found forFinite Element Model every harmonic term from k = 0 to N. Limitations of the present SAC program include the Next, the nodes of all ring elements are condensed [23] to leaveassumptions of geometrical and load symmetry about the cable axis and only two nodal points, A and B, on the inner and outer surfaces of thematerial isotropy. If dissimilar components are arranged within a cable component as shown in Figure 3. This leads to the stiffness matrix,core as illustrated in Figure 4, the radial deformation of the core will [k], of the condensed ring element,not be uniform about the cable axis. Also, fiber-reinforced layers usedfor hydraulic tubes and protective jackets may possess sufficientanisotropy to influence cable deformations. A new finite elementmodel has been developed that accounts for material anisotropy andunsymmetrical radial deformations. The example presented in thefollowing section illustrates the accuracy of this element. A briefdiscussion of this model is given below; however, a detailed derivationwill be presented in a forthcoming publication. This is an 8(N+1 square matrix with all harmonic terms decoupled In Briefly, each component in the cable core is modeled as a equation (20),macro-element having nodal degrees-of-freedom at all points of theinner and outer surfaces that contact adjacent components; for )example, points C, D and E in Figure 3. Ring Element. Due to the unsymmetrical deformations These matrices correspond to the nodal displacement vectors,expected for a typical ring element, radial and circumferentialdisplacement shape functions, u(r,8) and v(r,$), respectively, areexpressed in polar coordinates according to the following Fourier series[2] with quadratic variations in the radial direction: N Macro-Element. The condensed ring element that represents the N component in Figure 3 can now be treated as a linear elastic structure v(r,8) = [ v,(r)sink8 - va(r) cosk8 ] (18) in static equilibrium with the applied forces applied at its contact k=O points C, D and E. The unit displacement theorem [19] is used to obtain the stiffness matrix, [k],, of this component with respect to the U and v displacements at all contact points. 1324
- 7. A column vector of [k],, corresponding to a unit displacement where [K] is the stiffness matrix for all cable components; (Q) is the(U or v) at any contact point, is generated by determining the attendant unknown global displacement vector of all component contact points;reaction forces for the degrees-of-freedom at all contact points. A and (R) is the global load vector consisting of concentrated loadstypical contact point, C, making an angle, Bc, with the axis line, 5 = 0 applied at the contact points.as shown in Figure 3, illustrates the procedure. If a unit displacement,u=l, is imposed at point C, equations (17) and (18) become, u(r,5) = 1 Prototype Testing (5 = &; r = rB ) = 0 (5 = BD, BE; r = r, ) (23) The final verification of the cable design is to subject the as- v(r,O) = 0 (5 = Bc. BD, BE; r =rg 1 built cable to simultaneous conductor and mechanical performance tests. These tests are needed to confirm that all conductors perform satisfactorily while all pertinent proof loads are applied to the cable.where rB = outer radius of the component. Substituting comparable Measured mechanical data may include cable strain, torque or rotation,equations for all contact points leads to the displacement constraint flexural rigidity, cable diametrical reduction and cable breakingequation, strength. Analytical models that accurately predict the fatigue strength of a cable have not yet been developed, although a modest attempt has been made to describe the cyclical tension fatigue mechanism [SI. Thewhere [p] consists of constant terms related to the j angles above; (9) 3 most important failure mode is flexural fatigue where radiallyis the unknown displacement vector, compressed layers experience relative motion during cable bending. This motion may cause surface fatigue cracks (fretting) to develop at interlayer contact points. Cables which are to be operated over shipboard winches and sheaves should be subjected to rigorous cyclical bending tests. DESIGN EXAMPLEwhere (q),,Lk) are expressed by equation (22). (rp) is a column vectorwhose elements equal 0 or 1. The remotely-operated vehicle (ROV) electro-optical umbilical cable depicted in Figure 4 will be subjected to the design process A penalty function will be defined as described in Figure 1. Three electrical conductors (layer 2) and three jacketed optical fibers (layer 3 ) are contained within a core jacket (layer 5). Filler wires (layers 1 and 4) serve to stiffen the core package. Two round wire layers provide the external armor (layers 6 and 7). All core components are served with a common lay length and the twosuch that (t) = (0) when equation (24) is satisfied. The total potentialenygy, II, of the structure can be augmented by the quantity, armor layers are contrahelically laid to reduce the induced torque.(t) [c](t)/2, without changing its value as (t) approaches ( ) [c] is a 0.diagonal matrix of constant penalty parameters, ci. According to theprinciple of minimum potential energy, equilibrium is achieved for the Preliminary Designstationary condition, aIT/aq, = 0, which leads to the result, The electrical, optical and filler wires are placed in the most [ WI, + [ P I ~ M [ P]I41 = [PI~[cI{~,} I (27) compact form possible and contained within a high-density polyethylene sheath having a 11.15 mm outer diameter. This cable is to be subjected to a 68 kN tensile load and a factor of safety on yieldwhere [k], is given by equation (20). of at least 2.5 is desired. The armor wires are super extra-improved galvanized plow steel with an elastic modulus, E = 1 . 9 3 ~ 1 0 ~ and MPa, yield strength, S, = 1.97~10 Mpa. Following the penalty method [2][24], the parameters, ci, areincrementally determined such that (9) found by equation (27) satisfiesequation (26); viz., (t)=(O). Having determined (q), the reaction forces Armor Area. The nomograph in Figure 2 is used to determineat any of the contact points can be found by the equations, the armor wire area needed for each layer. See Table 8 for a description of the procedure. For FS = 2, E/S = 100, and an initial guess of an 18 degree lay angle for each layer, d= 225 or E, = 0.44% is obtained. For two armor layers, nL = 2, the nomograph yields AE/T ( i = l at r=rA; i = 3 at r = r B ) = 130. Substituting the value for E and T given above, the armor area (28) for each layer, A = 46 mm2. N F,(r,B) = k=0 [ [hj]!k)(q}!k)sink5 - [hj]r){q}?)coskB ] Next we choose two standard round wire diameters of 1.93 mm and 1.27 mm for layers 6 and 7, respectively. Using equation (l), we ( i = 2 at r=rA; i = 4 at r = r B ) obtain 15 wires for layer 6 and 36 wires for layer 7.and j=1,4 for both equations. [kij],,.(k) and (q), Jk) are defined byequations (21) and (22), respectively. These ieaction forces are Wire Coverage. Coverage for both layers should beevaluated at all contact points. This solution represents one column of approximately 95%. From Figure 4, the pitch radii are found as R = ,the stiffness matrix, [k],, corresponding to the direction of the 6.54 mm and R = 8.14 mm. Equation (2) requires that the number of ,imposed unit-displacement. This process is repeated for all contact wires in layer 6 be increased to 19 wires.points. In this manner, the entire stiffness matrix for the macro-element of the component shown in Figure 3, [k],, is generated. Torque and Stress Balance. For this preliminary design, the Assembly Procedure. Finally, the stiffness matrices for all torque contribution of the core components is neglected. Selecting themacro-elements are transformed to a global coordinate system and outer layer 7 to be the dependent layer and the lay angle of layer 6 setassembled into the system equation for the entire cable cross-section to 18 degrees (right-hand lay), equation (3) yields -17.6 degrees for[I19 layer 7. This result is rounded to -18 degrees (left-hand lay). [KHQ)= {RI (29) 1325
- 8. CBBLE CROSS-SECTION ROU U m b i l i c a l C a b l e Layer nnez- D i a muter D i n Cnnl C nnl ~ .e .694E-91 .887E-E1 .246E+EE .12iE+9C .246E+99 .183E+BC .246E+90 .246E+BC .l l S E + B l .115E+91 .591E+91 .501E+01 .755E+91 P n e s s ENTER t o C o n t i n u e . . B n r l y s i r by t k c S B C C o m p u t e r P p o g r a n Figure 4. Design Example 70 70 60 60 A Mean Data SAC-z5 50 40 50 40 1 SAC/FEME(3[1 I 30 30$E3 20 Test D a t a 20 U . ... m 10 SAC 10 A U T, 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 17.1 17.2 17.3 17.4 17.5 17.6 17.7 (%) Axial Strain [mm/mm] C a b l e Diameter [mm] Figure 5. Axial Strain of ROV Umbilical Cable Figure 6. Diametrical Reduction of ROV Umbilical Cable 13%
- 9. Linear Analysis. Using the preliminary design values obtained [4] Durelli, A.J., Machida, S and Parks, V.S., "Strains and .above, a linear analysis is used to estimate the cable strain and torque. Displacements in a Steel Wire Strand," Naval Engineers Journal, Dec.,The effective axial rigidity of the core, considering only the copper 1972, pp. 85-90.conductors and polyethylene sheath, ACE, = 2 . 6 ~ 1 0 N. Also, the coreis approximated as an incompressible material (9 = 1). Equation (4), [5] Greenhill, A.G., Proc. Inst. Mech. Eng., London, 1883.reduced for a straight cable with fixed ends (Ad=O) becomes, [6] Knapp, R.H., "Derivation of a New Stiffness Matrix for Helically Armoured Cables Considering Tension and Torsion," Int. J . for Numerical Methods in Engineering, Vol. 14, 1979, pp. 515-529. [7] Knapp, R.H., "Torque Balance Design for Helically Armored Cables," ASME J. o f Engineering f o r Industry, Vol. 83, Feb., 1981, pp.from which E, = 0.40% (approximately the same value obtained using 61-66.the nomograph) and S = -2.28 N-m. This is an acceptably smalltorque. [8] Knapp, R.H. and Chiu, E., "Tension Fatigue Model for Helically Armored Cables," ASME J . o f Energy Resources Technology, Vol. 110, Mar. 1988. HelicalWire Axial Stress. The wire axial stresses computed withequation (12) yield a stress of approximately 670 MPa for both layers [9] Knapp, R.H., "Structural Modeling of Undersea Cables," ASME J .6 and 7. For a yield strength of 1,970 MPa, this gives an acceptable of Offshore Mechanics and Arctic Engineering, Vol. 1 11, Nov., 1989,factor of safety of 2.9. pp. 323-330. [IO] Knapp, R.H., "Users Guide to Knapp-SACTU Stress Analysis forComputer-Aided Design of Cables," Knapp Engineering, Inc., Aiea, HI, 1991. The SAC program was used to finalize this ROV umbilical [I I ] Lanteigne, J., "Theoretical Estimation of the Response of Helicallycable design. The SAC analysis considered all core and armor wire Armored Cables to Tension, Torsion and Bending," ASME J . ofcomponents illustrated in Figure 4. The nonlinear stress-strain curve Applied Mechanics, Vol. 52, June, 1985, pp. 423-432.for the polyethylene core sheath and indentation of the wire layer 6into the sheath layer 5 were considered in the analysis. [I21 Leonard, J.W., Tension Structures, McGraw-Hill, 1988, pg. 74. Following its manufacture, this cable was subjected to a tension [I31 Liu, F.C., "Kink Formation and Rotational Response of Single andtest where both the cable diameter and axial strain were measured. Multistrand Electromechanical Cables," Proc. Civil Engineering in theBased on the as-built geometry and a measured compressive stress- Oceans I I I , Vol. 1, June 9-12, 1975, pp. 546-568.strain curve for the polyethylene sheath, the SAC analysis wasperformed. Comparisons of this analysis with the experimental data [I41 McNamara, J.F., Obrien, P.J. and Gilroy, S.G., "Nonlinearare presented in Figures 5 and 6. Also shown in Figure 6 is the SAC Analysis of Flexible Risers Using Hybrid Finite Elements," ASMEanalysis enhanced with the finite element model of the radial core Proc. 5th Intl. Conf. on Offshore Mech. and Arctic Engr., Vol. 3,deformations described above. These analysis results are in excellent Tokyo, Japan, 1986, pp. 371-377.agreement with the experimental results. The SAC/FEM analysis wasable to more accurately predict the change of cable diameter as shown [I51 Nowak, G., "Computer Design of Electromechanical Cables forin Figure 6. Ocean Applications," Proc. IOth Annual MTS Conf., Washington, D.C., 1974, pp. 293-305. CONCLUSIONS [ 161 Obrien, T., "General Solution of Suspended Cable Problems," ASCE J. o f the Structural Division, Vol. 93, No. STI, 1967, pp. 1-26. This paper presents a design methodology for underseaumbilical cables. Simple analysis procedures to obtain preliminary 1171 Oliveira, J.G., Goto, Y. and Okamoto, T., "Theoretical anddesign estimates have been described. Cable design software and a new Methodological Approaches to Flexible Pipe Design and Application,"finite element have been introduced as a cable design tool. An example 17th Annual Offshore Tech. Conf., paper #5021, Houston, Texas, Mayof an as-built ROV umbilical cable demonstrates the design process 6-9, 1985, pp. 517-526.and verifies the accuracy of the computer software. [I81 Peyrot, A.H., "Marine Cable Structures," ASCE J . o f t h e Structural Division, Vol. 106, No. 12, 1980, pp. 2391-2403. ACKNOWLEDGEMENTS [I91 Przemieniecki, J.S., Theory of Matrix Structural Analysis, This work was supported in part by a grant from the U.S. McGraw-Hill, New York, 1968.Bureau of Mines, Marine Mineral Technology Center. The test resultsof the ROV umbilical cable were obtained by the first author at [20] Riggs, H.R. and Leraand, T., "Efficient Static Analysis and DesignSumitomo Electric Industries, Ltd. in Osaka, Japan. The authors are of Flexible Risers," Proc. 9th Intl. Conf. Offshore Mech. and Arcticgrateful to Dr. S. Takeuchi, Mr. Y. Murata, and Mr. T. Mitsui of Engr., Vol. IB, 1990, pp. 371-377.Sumitomo for their generous support of the cable tests. [21] Ross, A.L., "Cable Kinking Analysis and Prevention," Trans. ASME, Vol. 99, Feb., 1977, pg. 112. REFERENCES [22] Walton, T.S. and Polachek, H., "Calculation of Nonlinear Transient Motion of Cables," Applied Mathematics Dept., Report No. 1279,[I] Bathe, K.J. and Wilson, E.L., Numerical Methods in Finite Element David Taylor Model Basin, July, 1959.Analysis, Prentice-Hall, 1976. [23] Wilson, E.L., "The Static Condensation Algorithm," Inll. J. o f Num.[2] Cook, R.D., Concepts and Applications of Finite Element Analysis, Meth. in Engr., Vol, 8 , No.], 1974, pp. 198-203.2nd Ed., John Wiley & Sons, Inc., 1974. 1241 Zienkiewicz, O.C., The Finite Element Method, McGraw-Hill, 3rd[3] Coyne, J., "Analysis of the Formation and Elimination of Loops in Ed., 1977.Twisted Cables," IEEE J. o f Oceanic Engr., Vol. 15, No. 2, April, 1990,pp. 72-83. 1327

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