1. CRS2 - Double symmetric cross section 1.1. Data group identifier NEW COMPonent CRS2 1.2. Component type identifier CMPTYP-ID TEMP CMPTYP-ID: character(8): Component type identifier TEMP: real: Temperature at which the specification applies Dummy in present version Figure 1. Cross section with 2 symmetry planes 1.3. Mass and volume AMS AE AI RGYR AMS: real: Mass per unit length \(\mathrm {[M/L]}\) AE: real: External cross-sectional area \(\mathrm {[L^2]}\) AI: real: Internal cross-sectional area \(\mathrm {[L^2]}\) RGYR: real: Radius of gyration about local x-axis \(\mathrm {[L]}\) AE is used to calculate buoyancy. AI is used to calculate additional mass of internal fluid if present. Otherwise AI is dummy. Note that the mass center is located along the local X-axis, i.e. at the origin of the local Y- and Z-axes. 1.4. Stiffness properties classification IEA IEJ IGT IPRESS IEA: integer, default: 0: Axial stiffness code 1 - constant stiffness N - table with N pairs of tension-elongation to be specified N >= 2 IEJ: integer, default: 0: 0 - zero bending stiffness 1 - constant stiffness N - table with N pairs of bending moment - curvature to be specified. N >= 2 IGT: integer, default: 0: Torsion stiffness code 0 - zero torsional stiffness 1 - constant stiffness -1- non-symmetric ``constant'' stiffness N - symmetric, N (positive) pairs specified -N- general torsion/relation (non-symmetric) N pairs specified N >= 2 IPRESS: integer, default: 0: Pressure dependency parameter related to bending moment 0 - no pressure dependency 1 - linear dependency (not implemented) NP - NP sets of stiffness properties to be given, corresponding to a table of NP pressure values (not implemented) 2 ⇐ NP ⇐ 10 Normally IEJ and IGT should both be zero or both greater than zero to assure stability in the FEM analysis. IPRESS=0 in this version of the program. 1.5. Bending-torsion geometric coupling specification This data group is optional, and will only be applied for IEJ=1 and IGT=1. BTGC BTGC: character(4): bending-torsion coupling identifier. If the BTGC identifier is present, geometric coupling between torsion and bending is accounted for. 1.6. Axial stiffness. Case 1 IEA=1 EA EA: real > 0: Axial stiffness \(\mathrm {[F]}\) 1.7. Axial stiffness. Case 2 IEA=N EAF(1) ELONG(1) . . . EAF(N) ELONG(N) EAF(1): real: Axial force corresponding to relative elongation ELONG(1) \(\mathrm {[F]}\) ELONG(1): real: Relative elongation () . . . EAF(N): real: ELONG(N): real: The pairs of EAF and ELONG must be given in increasing order. See also the figure Axial force corresponding to relative elongation. 1.8. Bending stiffness properties The amount of input depends upon the parameters IEJ and IPRESS according to the table below: Case: 0, IEJ: 0, IPRESS: 0, Data required: None. Case: 1, IEJ: 1, IPRESS: 0, Data required: EJY, EZJ, MFY, MF2. Case: 2, IEJ: 1, IPRESS: 1, Data required: Not implemented. Case: 3, IEJ: N, IPRESS: 0, Data required: CURV(I): I=1,N. BMOMY(I): I=1,N. BMOMZ(I) Case: 4, IEJ: N1, IPRESS: N2, Data required: Not implemented. Thus, the following data are required for the respective cases: 1.9. Bending stiffness. Case 1, IEJ=1 IPRESS=0 EJY EJZ GAsZ GAsY EJY: real > 0: Bending stiffness around local y-axis \(\mathrm {[FL^2]}\) EJZ: real > 0: Bending stiffness around z-axis \(\mathrm {[FL^2]}\) GAsZ: real: Shear stiffness in Z-direction \(\mathrm {[F]}\) GAsY: real: Shear stiffness in Y-direction \(\mathrm {[F]}\) The shear stiffness, GAsZ and GAsY, are optional input parameters. Specified GAsZ>0 and GAsY>0 will include shear deformation. This requires that all stiffness properties are constant, i.e. IEA = 1, IEJ = 1, IGT = 1. Note that the shear center is located along the local X-axis, i.e. at the origin of the local Y- and Z-axes. 1.10. Bending stiffness. Case 2, IEJ=1 IPRESS=1 (Not implemented) EJY(1) EJZ(1) PRESS(1) EJY(2) EJZ(2) PRESS (2) EJY(1): real: Bending stiffness around local y-axis \(\mathrm {[FL^2]}\) EJZ(1): real: Bending stiffness around local z-axis \(\mathrm {[FL^2]}\) PRESS(1): real: Pressure at which the above values apply \(\mathrm {[F/L^2]}\) EJY(2): real: Bending moments corresponding to 2nd pressure level, see description above EJZ(2): real: PRESS(2): real: PRESS(1) < PRESS(2) Figure 2. Bending stiffness around y-axis as function of pressure. Values at other pressure levels than PRESS(1) and PRESS(2) are obtained by linear interpolation/ extrapolation. 1.11. Bending stiffness description. Case 3 IEJ=N IPRESS=0 This specification consists of three different input lines. Curvature CURV(1) ... CURV(N) CURV(1): real: Curvature values for which bending moment is specified \(\mathrm {[1/L]}\) . . . CURV(N): real: To be specified in increasing order CURV=1/CURVATURE RADIUS Bending moment, y-axis BMOMY(1) . . . BMOMY(N) BMOMY(1): real: Bending moment around local y-axis \(\mathrm {[FL]}\) . . . BMOMY(N): real Bending moment, z-axis BMOMZ(1) . . . BMOMZ(N) BMOMZ(1): real: Bending moment around local z-axis \(\mathrm {[FL]}\) . . . BMOMZ(N): real CURV(1), BMOMY(1) and BMOMZ(1) have to be zero. See also the figure Bending moment around y-axis as function of curvature. 1.12. Bending stiffness. Case 4 IEJ=N1, IPRESS=N2 (Not implemented) This specification consists of four different input lines. Curvature CURV(1) ... CURV(N) CURV(1): real: Curvature values for which bending moments are specified \(\mathrm {[1/L]}\) . . . CURV(N): real: To be specified in increasing order CURV=1/CURVATURE RADIUS CURV(1) has to be zero. See also the figure Bending moment around y-axis as function of curvature. Pressure PRESS(1) ... PRESS(N) PRESS(1): real: Pressure levels for which bending moments are specified \(\mathrm {[F/L^2]}\) . . . PRESS(N): real: Bending moment, y-axis BMOMY(1,1) . . . BMOMY(N1,N2) BMOMY(I,J): real: Bending moment about local y-axis at curvature I and pressure J \(\mathrm {[FL]}\). . . . BMOMY(N1,N2):real: BMOMY(1,J), J=1, N2 have to be zero. Bending moment, z-axis BMOMZ(1,1) . . . BMOMZ(N1,N2) BMOMZ(I,J): real: Bending moment about local Z-axis at curvature I and pressure J \(\mathrm {[FL]}\). . . . BMOMZ(N1,N2):real: BMOMZ(1,J), J=1, N2 have to be zero. 1.13. Torsion stiffness Constant torsion stiffness. Case 1 |IGT|=1 GT- GT+ GT-: real > 0: Torsion stiffness (negative twist) \(\mathrm {[FL^2/Radian]}\) GT+: real: D.o. for positive twist. Dummy for IGT=1 Nonlinear torsion stiffness. Case 2 |IGT|= N TMOM(1) TROT(1) . . . TMOM(N) TROT(N) TMOM(1): real: Torsion moment \(\mathrm {[FL]}\) TROT(1): real: Torsion angle/length \(\mathrm {[Radian/L]}\) If IGT is positive TMOM(1) and TROT(1) have to be zero. TROT must be given in increasing order. 1.14. Damping specification Identical to input for cross-section type CRS1, see data group Damping specification. 1.15. Hydrodynamic load type identification, One input line CHLOAD CHLOAD: character: = HYDR - Text to identify hydrodynamic load type Note: Required if non-Morison loads are to be specified Load type identification for CHLOAD=HYDR, One input line CHTYPE CHTYPE: character: Hydrodynamic load type = NONE: No hydrodynamic load = MORI: Load based on Morisons generalized equation. Sea surface penetration formulation = MORP: As MORI, but improved by taking into account partially submerged cross-section = MACF: Load based on MacCamy-Fuchs potential equations and quadratic drag load = POTN: Load based on input of force transfer functions and retardation fuctions from 3rd party programs and quadratic drag load (Under development) Note that the option POTN currently is under testing. Potential flow forces are only available for irregular time domain analysis. Hydrodynamic force coefficients if CHTYPE=MORI or CHTYPE=MORP, submerged cross section CHTYPE=MORP is similar to CHTYPE=MORI but with three key differences: - the load calculated at a cross-section is reduced in proportional with the instantaneous wet portion of the cross-section. - the Froude-Krylov term used longitudinal direction in Morison’s equation is replaced by the product of the dynamic pressure and the submerged area at each end of the element. The external area for this purpose is assumed to be circular. - If the specified value for external area (AE) is zero, neither hydrostatic nor hydrodynamic loads will act on the cross section. Definitions of dimensional hydrodynamic force coefficients for a fully submerged cross section are given below CDX CDY CDZ CDTMOM AMX AMY AMZ AMTOR CDLX CDLY CDLZ SCFKN SCFKT CDX: real: Drag force coefficient for local x-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDY: real: Drag force coefficient for local y-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDZ: real: Drag force coefficient for local z-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDTMOM: real: Drag force coefficient for local x-rotation. Not used in present version. AMX: real: Added mass per length in x-direction \(\mathrm {[M/L]}\) AMY: real: Added mass per length in y-direction \(\mathrm {[M/L]}\) AMZ: real: Added mass per length in z-direction \(\mathrm {[M/L]}\) AMTOR: real: Added mass for local x-rotation \(\mathrm {[ML^2/L]}\) Not used in present version. CDLX: real, default: 0: Linear drag force coefficients in local x-direction \(\mathrm {[F/((L/T)\times L)]}\) CDLY: real, default: 0: Linear drag force coefficients in local y-direction \(\mathrm {[F/((L/T)\times L)]}\) CDLZ: real, default: 0: Linear drag force coefficients in local z-direction \(\mathrm {[F/((L/T)\times L)]}\) SCFKN: real, default: 1: Scaling factor for the Froude-Krylov term in Morison’s equation in normal direction SCFKT: real, default: 1: Scaling factor for the Froude-Krylov term in Morison’s equation in tangential direction. Only the values 0.0 and 1.0 are permitted. The drag forces per unit length acting in the local coordinate system are computed as: \(\mathrm {F_x=CDX\times VRELX\times VRELX+CDLX\times VRELX}\) \(\mathrm {F_y=CDY\times \sqrt{VRELY^2+VRELZ^2}\times VRELY+CDLY\times VRELY}\) \(\mathrm {F_z=CDZ\times \sqrt{VRELY^2+VRELZ^2}\times VRELZ+CDLZ\times VRELZ}\) where: \(\mathrm {CDX,CDY,CDZ}\): are the input quadratic drag force coefficients in local x, y and z-directions \(\mathrm {CDLX,CDLY,CDLZ}\): are the input linear drag force oefficients in local x, y and z-directions \(\mathrm {VRELX,VRELY,VRELZ}\): are relative water velocities in local x, y and z-directions The input quadratic drag force coefficients \(\mathrm {CDX}\), \(\mathrm {CDY}\) and \(\mathrm {CDZ}\) will normally be calculated as: \(\mathrm {CDX=\frac{1}{2}\rho S_{2D}C_{dx}}\) \(\mathrm {CDY=\frac{1}{2}\rho B_yC_{dy}}\) \(\mathrm {CDZ=\frac{1}{2}\rho B_zC_{dz}}\) where: \(\mathrm {\rho }\): water density \(\mathrm {S_{2D}}\): cross sectional wetted surface \(\mathrm {B_y,B_z}\): projected area per. unit length for flow in local y and z-direction, respectively \(\mathrm {C_{dx},C_{dy},C_{dz}}\): nondimensional drag coefficients in local x, y and z-directions, respectively The input added mass per. unit length \(\mathrm {AMX}\), \(\mathrm {AMY}\) and \(\mathrm {AMZ}\) can be calculated as: \(\mathrm {AMX=\rho AC_{mx}}\) \(\mathrm {AMY=\rho AC_{my}}\) \(\mathrm {AMZ=\rho AC_{mz}}\) where: * \(\mathrm {\rho }\): water density * \(\mathrm {A}\): cross sectional area * \(\mathrm {C_{mx},C_{my},C_{mz}}\): nondimensional added mass coefficients in local x, y and z-directions, respectively Hydrodynamic force coefficients if CHTYPE=MACF MacCamy-Fuchs frequency-dependent hydrodynamic loads on a stationary vertical circular cylinder will be applied for CHTYPE=MACF. MacCamy-Fuchs forces are pre-computed based on the element position after static calculation. MacCamy-Fuchs forces are only available for irregular time domain analysis. Quadratic drag may also be applied on elements with MacCamy-Fuchs loading. McCamy Fuchs assumes that the cross-section is circular, so a single transverse quadratic drag coefficient is given (CDZ will be set to CDY). CQX CQY ICODE D CQX: real: Quadratic drag coefficient in tangential direction ICODE=1: CQX=CDX: dimensional drag force coefficient \(\mathrm {[F/((L/T)^2\times L)]}\) ICODE=2: CQX=Cdt: nondimensional drag force coefficient CQY: real: Quadratic drag coefficient in normal direction ICODE=1: CQY=CDY: dimensional drag force coefficient \(\mathrm {[F/((L/T)^2\times L)]}\) ICODE=2: CQY=Cdn: nondimensional drag force coefficient ICODE: integer: Code for input of hydrodynamic drag coefficients ICODE=1: Dimensional coefficients ICODE=2: Nondimensional coefficients D: real, default:\(\sqrt{\mathrm {\frac{4}{\pi }(AE)}}\): Hydrodynamic diameter of the pipe \(\mathrm {[L]}\). Default value is calculated from external cross-sectional area given as input in data section Mass and volume Hydrodynamic force coefficients if CHTYPE=POTN Frequency-dependent added mass, radiation damping, and excitation forces based on the first order potential flow solution will be applied for CHTYPE=POTN. The radiation and diffraction coefficients are to be given by a separate input file specified under the data group Potential flow library specification. Quadratic drag may also be applied on cross-sections with potential flow loading. CQX CQY CQZ ICODE D SCFKT CQX: real: Quadratic drag coefficient in local x-direction ICODE=1: CQX=CDX: dimensional drag force coefficient \(\mathrm {[F/((L/T)^2\times L)]}\) ICODE=2: CQX=Cdt: nondimensional drag force coefficient CQY: real: Quadratic drag coefficient in local y-direction ICODE=1: CQY=CDY: dimensional drag force coefficient \(\mathrm {[F/((L/T)^2\times L)]}\) ICODE=2: CQY=Cdn: nondimensional drag force coefficient CQZ: real: Quadratic drag coefficient in local z-direction ICODE=1: CQZ=CDZ: dimensional drag force coefficient \(\mathrm {[F/((L/T)^2\times L)]}\) ICODE=2: CQZ=Cdn: nondimensional drag force coefficient ICODE: integer, default: 1: ICODE Code for input of hydrodynamic force coefficients ICODE=1: Dimensional coefficients ICODE=2: Nondimensional coefficients D: real, default:\(\sqrt{\mathrm {\frac{4}{\pi }(AE)}}\): Hydrodynamic diameter \(\mathrm {[L]}\). Default value is calculated from external cross-sectional area given as input in data section Mass and volume SCFKT: real, default: 1: Scaling factor for the Froude-Krylov term in Morison’s equation in tangential direction. Only the values 0.0 and 1.0 are permitted. 1.16. Aerodynamic load type identification, One optional input line CHLOAD CHLOAD: character: = WIND - Text to identify wind coefficients 1.17. Load type identification, One optional input line CHTYPE CHTYPE: character: Type of wind load coefficients = MORI: Morison-like loading, Drag term = AIRC: Air foil cross section to be specified (Not implemented) = AIRF: Air foil cross section, Refers to a air foil library file CHTYPE=MORI: Morison-like aerodynamic drag, One input line CDXAERO CDYAERO CDZAERO CDXAERO: real: Dimensional quadratic drag coefficient for local x-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDYAERO: real: Dimensional quadratic drag coefficient for local y-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDZAERO: real: Dimensional quadratic drag coefficient for local z-direction \(\mathrm {[F/((L/T)^2\times L)]}\) The drag forces per unit length acting in the local coordinate system are computed as: - \(\mathrm {F_x=CDXAERO\times VRELX\times |VRELX|}\) - \(\mathrm {F_y=CDYAERO\times VRELY\times \sqrt{VRELY^2+VRELZ^2}}\) - \(\mathrm {F_z=CDZAERO\times VRELZ\times \sqrt{VRELY^2+VRELZ^2}}\) where: - \(\mathrm {CDXAERO,CDYAERO,CDZAERO}\): are the input quadratic drag force coefficients in local x, y and z-directions - \(\mathrm {VRELX,VRELY,VRELZ}\): are relative wind velocities in local x, y and z-directions The input quadratic drag force coefficients \(\mathrm {CDX}\), \(\mathrm {CDY}\) and \(\mathrm {CDZ}\) will normally be calculated as: - \(\mathrm {CDXAERO=\frac{1}{2}\rho _{air}S_{2D}C_{dx}}\) - \(\mathrm {CDYAERO=\frac{1}{2}\rho _{air}B_yC_{dy}}\) - \(\mathrm {CDZAERO=\frac{1}{2}\rho _{air}B_zC_{dz}}\) where: - \(\mathrm {\rho _{air}}\): air density - \(\mathrm {S_{2D}}\): cross sectional surface area - \(\mathrm {B_y,B_z}\): projected area per. unit length for flow in local y and z-direction, respectively - \(\mathrm {C_{dx},C_{dy},C_{dz}}\): nondimensional drag coefficients in local x, y and z-directions, respectively If the component is part of a wind turbine tower line, only the CDY component is used for tower shadow computation. CHTYPE=AIRF: Coefficients on file. ID and chord length, One input line CHCOEF CHORDL YFC ZFC ROTFAX CHCOEF: character(32): Air foil coefficient identifier. Must be found on the air foil library file CHORDL: real: Chord length of foil section. \(\mathrm {[L]}\) It is used to scale the air foil load coefficients. YFC: real, default: 0: Y-coordinate of foil origin \(\mathrm {[L]}\) ZFC: real, default: 0: Z-coordinate of foil origin \(\mathrm {[L]}\) ROTFAX: real, default: 0: Inclination of foil system \(\mathrm {[deg]}\) The blade coordinate system and origin coincides with the elastic (local) \(\mathrm {(X_L,Y_L,Z_L)}\) coordinate system. The aerodynamic coordinate system \(\mathrm {(X_{AF},Y_{AF})}\) is located at (YFC,ZFC) referred to the local coordinate system, and is rotated about the blade x axis by the angle ROTFAX, as indicated in the figure below. The \(\mathrm {X_L}\) axis is pointing into the paper plane, while the \(\mathrm {Z_{AF}}\) is pointing out of plane. Note that the air foil coefficients has to be referred to the aerodynamic coordinate system as indicated by the corresponding angle of attack in the figure. For airfoil elements that are part of a wind turbine blade, the local \(\mathrm {X_L}\)-axis is pointing towards the blade tip. Normally, the arodynamic twist and the structural twist are given as one input. The input is given as twist of the elastic local coordinate system (see Line and segment specification ). ROTFAX should normally be 0. Figure 3. Definition of foil center and inclination of foil system in the local cross section (strength). In coupled analysis, a SIMO wind type with IWITYP >= 10 must be used if the case contains elements with wind force coefficients that are not on the blades of a wind turbine. 1.18. Capacity parameter TB YCURMX ZCURMX TB: real: Tension capacity \(\mathrm {[F]}\) YCYRMX: real: Maximum curvature around local y-axis \(\mathrm {[1/L]}\) ZCURMX: real: Maximum curvature around local z-axis \(\mathrm {[1/L]}\) These parameters are dummy in the present version 2. CRS7 - General cross section 2.1. Data group identifier NEW COMPonent CRS7 2.2. Component type identifier CMPTYP-ID TEMP ALFA CMPTYP-ID: character(8): Component type identifier TEMP : real: Temperature at which the specification applies Dummy in present version ALPHA: real: Thermal expansion coefficient \(\mathrm {[Temp^{-1}]}\) Dummy in present version Figure 4. General cross-section 2.3. Mass YECC_MASS ZECC_MASS YECC_MASS: real: Mass center coordinate \(\mathrm {Y_m}\) in beam element system \(\mathrm {[L]}\) ZECC_MASS: real: Mass center coordinate \(\mathrm {Z_m}\) in beam element system \(\mathrm {[L]}\) AMS RGYR AMS : real: Mass per unit length \(\mathrm {[M/L]}\) RGYR: real: Radius of gyration about mass center \(\mathrm {(Y_m,Z_m)}\) \(\mathrm {[L]}\) 2.4. Buoyancy YECC_BUOY ZECC_BUOY YECC_BUOY: real: Buoyancy center Y-coordinate in beam element system \(\mathrm {[L]}\) Dummy in present version. Bouyancy center set equal to mass center. ZECC_BUOY: real: Buoyancy center Z-coordinate in beam element system \(\mathrm {[L]}\) Dummy in present version. Bouyancy center set equal to mass center. AE AI AE: real: External cross-sectional area \(\mathrm {[L^2]}\) Basis for calculation of buoyancy AI: real: Internal cross-sectional area \(\mathrm {[L^2]}\) Dummy in present version 2.5. Stiffness properties Only constant stiffness properties are allowed. 2.6. Area center and principal axes The area center is the cross-section point where the axial force acts through. The principal axes are formally determined from the requirement \(\int_AV\,W\,\,\mathrm {d}A=0\), where \(\mathrm {V}\) and \(\mathrm {W}\) denote the principal coordinates and \(\mathrm {A}\) is the cross-section area. The orientation of the principal axes is defined in terms of a positive X-rotation \(\mathrm {\theta }\) relative to the beam element YZ-coordinate system as shown in the figure General cross-section ~ YECC_AREACENT ZECC_AREACENT THETA ~ YECC_AREACENT: real: Area center coordinate \(\mathrm {Y_a}\) in beam element system \(\mathrm {[L]}\) ZECC_AREACENT: real: Area center coordinate \(\mathrm {Z_a}\) in beam element system \(\mathrm {[L]}\) THETA: real: Orientation \(\mathrm {\theta }\) of principal axes V and W [deg.]. See figure General cross-section. 2.7. Shear center The shear center represents the attack point of the shear forces. YECC_SHEARCENT ZECC_SHEARCENT YECC_SHEARCENT: real: Shear center coordinate \(\mathrm {Y_s}\) in beam element system \(\mathrm {[L]}\) ZECC_SHEARCENT: real: Shear center coordinate \(\mathrm {Z_s}\) in beam element system \(\mathrm {[L]}\) 2.8. Axial stiffness EA EA: real > 0: Axial stiffness \(\mathrm {[F]}\) 2.9. Bending stiffness The bending stiffness refers to the principal axes V and W, see figure General cross-section. EJV EJW EJV: real > 0: Bending stiffness about principal V-axis \(\mathrm {[FL^2]}\) EJW: real > 0: Bending stiffness about principal W-axis \(\mathrm {[FL^2]}\) 2.10. Shear stiffness The shear stiffness refers to the principal axes V and W, see figure General cross-section. GAsW GAsV GAsW: real: Shear stiffness in principal W-direction \(\mathrm {[F]}\) GAsV: real: Shear stiffness in principal V-direction \(\mathrm {[F]}\) The shear stiffness, GAsW and GAsV, are optional input parameters. Specified GAsW>0 and GAsV>0 will include shear deformation. 2.11. Torsion stiffness GT GT: real > 0: Torsion stiffness \(\mathrm {[FL^2/Radian]}\) For a circular cross-section the torsion stiffness is given by the polar moment of inertia. Note that this is not the case for non-circular cross-sections. 2.12. Bending-torsion geometric coupling This data group is optional. BTGC BTGC: character(4): bending-torsion coupling identifier. If the BTGC identifier is present, geometric coupling between torsion and bending is accounted for. 2.13. Damping specification Identical to input for cross-section type CRS1, see data group Damping specification. The stiffness matrix used as basis for the Rayleigh damping includes only the material stiffness matrix. The geometric stiffness matrix is not included as this would introduce damping of the rigid body motion for CRS7. 2.14. Hydrodynamic load type identification, One input line CHLOAD CHLOAD: character: = HYDR - Text to identify hydrodynamic coefficients Note: Required if non-Morison loads are to be specified Load type identification for CHLOAD=HYDR, One input line CHTYPE CHTYPE: character: Hydrodynamic load type = NONE: No hydrodynamic load coefficients = MORI: Slender element hydrodynamic coefficients Hydrodynamic force coefficients if CHTYPE=MORI CDX CDY CDZ CDTMOM AMX AMY AMZ AMTOR CDLX CDLY CDLZ SCFKN SCFKT CDX: real: Drag force coefficient for local x-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDY: real: Drag force coefficient for local y-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDZ: real: Drag force coefficient for local z-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDTMOM: real: Drag force coefficient for local x-rotation. Not used in present version. AMX: real: Added mass per length in x-direction \(\mathrm {[M/L]}\) AMY: real: Added mass per length in y-direction \(\mathrm {[M/L]}\) AMZ: real: Added mass per length in z-direction \(\mathrm {[M/L]}\) AMTOR: real: Added mass for local x-rotation \(\mathrm {[ML^2/L]}\) CDLX: real, default: 0: Linear drag force coefficients in local x-direction \(\mathrm {[F/((L/T)\times L)]}\) CDLY: real, default: 0: Linear drag force coefficients in local y-direction \(\mathrm {[F/((L/T)\times L)]}\) CDLZ: real, default: 0: Linear drag force coefficients in local z-direction \(\mathrm {[F/((L/T)\times L)]}\) SCFKN: real, default: 1: Scaling factor for the Froude-Krylov term in Morison’s equation in normal direction SCFKT: real, default: 1: Scaling factor for the Froude-Krylov term in Morison’s equation in tangential direction. Only the values 0.0 and 1.0 are permitted. The drag forces per unit length acting in the local coordinate system are computed as: \(\mathrm {F_x=CDX\times VRELX\times VRELX+CDLX\times VRELX}\) \(\mathrm {F_y=CDY\times VRELY\times VRELY+CDLY\times VRELY}\) \(\mathrm {F_z=CDZ\times VRELZ\times VRELZ+CDLZ\times VRELZ}\) where: \(\mathrm {CDX,CDY,CDZ}\): are the input quadratic drag force coefficients in local x, y and z-directions \(\mathrm {CDLX,CDLY,CDLZ}\): are the input linear drag force coefficients in local x, y and z-directions \(\mathrm {VRELX,VRELY,VRELZ}\): are relative water velocities in local x, y and z-directions The input quadratic drag force coefficients \(\mathrm {CDX}\), \(\mathrm {CDY}\) and \(\mathrm {CDZ}\) will normally be calculated as: \(\mathrm {CDX=\frac{1}{2}\rho S_{2D}C_{dx}}\) \(\mathrm {CDY=\frac{1}{2}\rho B_yC_{dy}}\) \(\mathrm {CDZ=\frac{1}{2}\rho B_zC_{dz}}\) where: \(\mathrm {\rho }\): water density \(\mathrm {S_{2D}}\): cross sectional wetted surface \(\mathrm {B_y,B_z}\): projected area per. unit length for flow in local y and z-direction, respectively \(\mathrm {C_{dx},C_{dy},C_{dz}}\): nondimensional drag coefficients in local x, y and z-directions, respectively The input added mass per. unit length \(\mathrm {AMX}\), \(\mathrm {AMY}\) and \(\mathrm {AMZ}\) can be calculated as: \(\mathrm {AMX=\rho AC_{mx}}\) \(\mathrm {AMY=\rho AC_{my}}\) \(\mathrm {AMZ=\rho AC_{mz}}\) where: * \(\mathrm {\rho }\): water density * \(\mathrm {A}\): cross sectional area * \(\mathrm {C_{mx},C_{my},C_{mz}}\): nondimensional added mass coefficients in local x, y and z-directions, respectively 2.15. Aerodynamic load type identification, One optional input line CHLOAD CHLOAD: character: = WIND - Text to identify wind coefficients 2.16. Load type identification, One optional input line CHTYPE CHTYPE: character: Type of wind load coefficients = MORI: Morison-like loading, Drag term = AIRC: Air foil cross section to be specified (Not implemented) = AIRF: Air foil cross section, Refers to a air foil library file CHTYPE=MORI: Morison-like aerodynamic drag, One input line CDXAERO CDYAERO CDZAERO CDXAERO: real: Dimensional quadratic drag coefficient for local x-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDYAERO: real: Dimensional quadratic drag coefficient for local y-direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDZAERO: real: Dimensional quadratic drag coefficient for local z-direction \(\mathrm {[F/((L/T)^2\times L)]}\) The drag forces per unit length acting in the local coordinate system are computed as: \(\mathrm {F_x=CDXAERO\times VRELX\times |VRELX|}\) \(\mathrm {F_y=CDYAERO\times VRELY\times \sqrt{VRELY^2+VRELZ^2}}\) \(\mathrm {F_z=CDZAERO\times VRELZ\times \sqrt{VRELY^2+VRELZ^2}}\) where: \(\mathrm {CDXAERO,CDYAERO,CDZAERO}\): are the input quadratic drag force coefficients in local x, y and z-directions \(\mathrm {VRELX,VRELY,VRELZ}\): are relative wind velocities in local x, y and z-directions The input quadratic drag force coefficients \(\mathrm {CDX}\), \(\mathrm {CDY}\) and \(\mathrm {CDZ}\) will normally be calculated as: \(\mathrm {CDXAERO=\frac{1}{2}\rho _{air}S_{2D}C_{dx}}\) \(\mathrm {CDYAERO=\frac{1}{2}\rho _{air}B_yC_{dy}}\) \(\mathrm {CDZAERO=\frac{1}{2}\rho _{air}B_zC_{dz}}\) where: \(\mathrm {\rho _{air}}\): air density \(\mathrm {S_{2D}}\): cross sectional surface area \(\mathrm {B_y,B_z}\): projected area per. unit length for flow in local y and z-direction, respectively \(\mathrm {C_{dx},C_{dy},C_{dz}}\): nondimensional drag coefficients in local x, y and z-directions, respectively If the component is part of a wind turbine tower line, only the CDY component is used for tower shadow computation. CHTYPE=AIRF: Coefficients on file. ID and chord length, One input line CHCOEF CHORDL YFC ZFC ROTFAX CHCOEF: character(32): Air foil coefficient identifier. Must be found on the air foil library file CHORDL: real: Chord length of foil section. \(\mathrm {[L]}\) It is used to scale the air foil load coefficients. YFC: real, default: 0: Y-coordinate of foil origin \(\mathrm {[L]}\) ZFC: real, default: 0: Z-coordinate of foil origin \(\mathrm {[L]}\) ROTFAX: real, default: 0: Inclination of foil system \(\mathrm {[deg]}\) The blade coordinate system and origin coincides with the elastic (local) \(\mathrm {(X_L,Y_L,Z_L)}\) coordinate system. The aerodynamic coordinate system \(\mathrm {(X_{AF},Y_{AF})}\) is located at (YFC,ZFC) referred to the local coordinate system, and is rotated about the blade x axis by the angle ROTFAX, as indicated in the figure below. The \(\mathrm {X_L}\) axis is pointing into the paper plane, while the \(\mathrm {Z_{AF}}\) is pointing out of plane. Note that the air foil coefficients has to be referred to the aerodynamic coordinate system as indicated by the corresponding angle of attack in the figure. For airfoil elements that are part of a wind turbine blade, the local \(\mathrm {X_L}\)-axis is pointing towards the blade tip. Note that suppliers of wind turbine blades normally give the foil twist relative to the the areodynamic coordinate system, i.e. as twist around the \(\mathrm {Z_{AF}}\) -axis. Definition of foil center and inclination of foil system in the local cross section (strength In coupled analysis, a SIMO wind type with IWITYP >= 10 must be used if the case contains elements with wind force coefficients that are not on the blades of a wind turbine. 2.17. Capacity parameter TB YCURMX ZCURMX TB: real: Tension capacity \(\mathrm {[F]}\) YCYRMX: real: Maximum curvature around local y-axis \(\mathrm {[1/L]}\) ZCURMX: real: Maximum curvature around local z-axis \(\mathrm {[1/L]}\) These parameters are dummy in the present version 3. BODY - Description of attached bodies 3.1. Data group identifier NEW COMPonent BODY 3.2. Component type identifier CMPTYP-ID CMPTYP-ID: character(8): Component type identifier A body is a component that may be attached at supernodes and segment interconnection points. The following essential properties should be observed: - The BODY is directly attached to a nodal point and has no motion degrees of freedom by itself. - The BODY component serves to add concentrated masses (inertia force), weight or buoyancy forces to the system. 3.3. Mass and volume AM AE AM: real: Mass \(\mathrm {[M]}\) AE: real: Displacement volume \(\mathrm {[L^3]}\) 3.4. Hydrodynamic coefficients ICOO CDX CDY CDZ AMX AMY AMZ ICOO: character(5): Coordinate system code ICOO=GLOBAL: Coefficients refer to global coordinate system ICOO=LOCAL: Coefficients refer to local coordinate system of neighbour elements in the actual line CDX: real: Drag force coefficient in X-direction \(\mathrm {[F/(L/T)^2)]}\) CDY: real: Drag force coefficient in Y-direction \(\mathrm {[F/(L/T)^2)]}\) CDZ: real: Drag force coefficient in Z-direction \(\mathrm {[F/(L/T)^2)]}\) AMX: real: Added mass in X-direction \(\mathrm {[M]}\) AMY: real: Added mass in Y-direction \(\mathrm {[M]}\) AMZ: real: Added mass in Z-direction \(\mathrm {[M]}\) The drag forces acting in the global/local coordinate system are computed as: \(\mathrm {F_x=CDX\times VRELX\times VRELX}\) \(\mathrm {F_y=CDY\times VRELY\times VRELY}\) \(\mathrm {F_z=CDZ\times VRELZ\times VRELZ}\) where: \(\mathrm {CDX,CDY,CDZ}\): are the input drag force coefficients in global/local x, y and z-directions, respectively \(\mathrm {VRELX,VRELY,VRELZ}\): are relative water velocities in global/local x, y and z-directions respectively The input quadratic drag force coefficients \(\mathrm {CDX}\), \(\mathrm {CDY}\) and \(\mathrm {CDZ}\) will normally be calculated as: \(\mathrm {CDX=\frac{1}{2}\rho B_xC_{dx}}\) \(\mathrm {CDY=\frac{1}{2}\rho B_yC_{dy}}\) \(\mathrm {CDZ=\frac{1}{2}\rho B_zC_{dz}}\) where: \(\mathrm {\rho }\): water density \(\mathrm {B_x,B_y,B_z}\): projected area for flow in global/local y and z-direction \(\mathrm {C_{dx},C_{dy},C_{dz}}\): nondimensional drag coefficients in global/local x, y and z-directions 4. CONB - Description of ball joint connectors This component can be used to model balljoint, hinges and universal joints. The component has zero length, and adds 6 degrees of freedom to the system model. The forces due to mass and weight are assumed to act at the nodal point at which the component is specified. Note that this component can not be used in branch lines in standard systems, or in combination with bar elements. Should also be used with care at supernodes with user defined boundary conditions for rotations in AR system to avoid singularities in the FEM solution procedure. 4.1. Data group identifier NEW COMPonent CONB 4.2. Component type identifier CMPTYP-ID CMPTYP-ID: character(8): Component type identifier 4.3. Mass and volume AM AE AM: real: Mass \(\mathrm {[M]}\) AE: real: Displacement volume \(\mathrm {[L^3]}\) 4.4. Hydrodynamic coefficients ICOO CDX CDY CDZ AMX AMY AMZ ICOO: character: Coordinate system code ICOO=GLOBAL: Coefficients refer to global coordinate system ICOO=LOCAL: Coefficients refer to local coordinate system of neighbour elements in the actual line CDX: real: Drag force coefficient in X-direction \(\mathrm {[F/(L/T)^2)]}\) CDY: real: Drag force coefficient in Y-direction \(\mathrm {[F/(L/T)^2)]}\) CDZ: real: Drag force coefficient in Z-direction \(\mathrm {[F/(L/T)^2)]}\) AMX: real: Added mass in X-direction \(\mathrm {[M]}\) AMY: real: Added mass in Y-direction \(\mathrm {[M]}\) AMZ: real: Added mass in Z-direction \(\mathrm {[M]}\) The drag forces acting in the global/local coordinate system are computed as: \(\mathrm {F_x=CDX\times VRELX\times VRELX}\) \(\mathrm {F_y=CDY\times VRELY\times VRELY}\) \(\mathrm {F_z=CDZ\times VRELZ\times VRELZ}\) where: \(\mathrm {CDX,CDY,CDZ}\): are the input drag force coefficients in global/local x, y and z-directions, respectively \(\mathrm {VRELX,VRELY,VRELZ}\): are relative water velocities in global/local x, y and z-directions respectively The input quadratic drag force coefficients \(\mathrm {CDX}\), \(\mathrm {CDY}\) and \(\mathrm {CDZ}\) will normally be calculated as: \(\mathrm {CDX=\frac{1}{2}\rho B_xC_{dx}}\) \(\mathrm {CDY=\frac{1}{2}\rho B_yC_{dy}}\) \(\mathrm {CDZ=\frac{1}{2}\rho B_zC_{dz}}\) where: \(\mathrm {\rho }\): water density \(\mathrm {B_x,B_y,B_z}\): projected area per. unit lengt for flow in global/local y and z-directions, respectively \(\mathrm {C_{dx},C_{dy},C_{dz}}\): nondimensional drag coefficients in global/local x, y and z-directions, respectively 4.5. Degrees of freedom IRX IRY IRZ IRX: integer, default: 0: Rotation freedom code, x-axis IRY: integer, default: 0: Rotation freedom code, y-axis IRZ: integer, default: 0: Rotation freedom code, z-axis 1 - Fixed (no deformation) 0 - Free (zero moment) x-, y- and z-axes refer to local coordinate system of the neighbour element in the line where the ball joint is specified. Figure 5. Rotation freedom for a ball joint component 5. FLEX - Description of flex-joint connectors This component can be used to model ball joints, hinges and universal joints with specified rotational stiffness. It will introduce one extra element with zero length at the segment end to which it is attached, and add 6 degrees of freedom to the system model. The translation dofs of freedom are suppressed by use of linear constraint equations. Note that this component can not be used in branch lines in standard systems, or in combination with bar elements. It should also be used with care at supernodes with user defined boundary conditions for rotations in AR system to avoid singularities in the FEM solution procedure. In present version, flex-joint connectors may only be used for nonlinear static and dynamic analysis. 5.1. Data group identifier NEW COMPonent FLEX 5.2. Component type identifier CMPTYP-ID CMPTYP-ID: character(8): Component type identifier 5.3. Mass and volume AM AE RGX RGY RGZ CRX CRY CRZ AM: real, default: 0: Mass \(\mathrm {[M]}\) AE: real, default: 0: Displacement volume \(\mathrm {[L^3]}\) RGX: real, default: 0: Radius of gyration around local x-axis \(\mathrm {[L]}\) RGY: real, default: 0: Radius of gyration around local y-axis \(\mathrm {[L]}\) RGZ: real, default: 0: Radius of gyration around local z-axis \(\mathrm {[L]}\) CRX: real, default: 0: Damping coeff. Rotational velocity around local x-axis \(\mathrm {[FLT/deg]}\) CRY: real, default: 0: Damping coeff. Rotational velocity around local y-axis \(\mathrm {[FLT/deg]}\) CRZ: real, default: 0: Damping coeff. Rotational velocity around local z-axis \(\mathrm {[FLT/deg]}\) 5.4. Hydrodynamic coefficients CDX CDY CDZ AMX AMY AMZ AMXROT AMYROT AMZROT CDX: real, default: 0: Drag coeff. in local x-direction \(\mathrm {[F/(L/T)^2)]}\) CDY: real, default: 0: Drag coeff. in local y-direction \(\mathrm {[F/(L/T)^2)]}\) CDZ: real, default: 0: Drag coeff. in local z-direction \(\mathrm {[F/(L/T)^2)]}\) AMX: real, default: 0: Added mass in local x-direction \(\mathrm {[M]}\) AMY: real, default: 0: Added mass in local y-direction \(\mathrm {[M]}\) AMZ: real, default: 0: Added mass in local z-direction \(\mathrm {[M]}\) AMXROT: real, default: 0: Added mass rotation around local x-direction \(\mathrm {[FL\times T^2]}\) AMYROT: real, default: 0: Added mass rotation around local y-direction \(\mathrm {[FL\times T^2]}\) AMZROT: real, default: 0: Added mass rotation around local z-direction \(\mathrm {[FL\times T^2]}\) The tangential drag force, the force acting in local x-axis, is computed by: \(\mathrm {FX=CDX\times VRELX\times |VRELX|}\) The drag force acting normal to the local x-direction, is assumed to act in the same direction as the relative velocity transverse component and are computed according to: \(\mathrm {FY=CDY\times \sqrt{VRELY^2+VRELZ^2}\times VRELY}\) \(\mathrm {FZ=CDY\times \sqrt{VRELY^2+VRELZ^2}\times VRELZ}\) 5.5. Stiffness properties classification IDOF IBOUND RAYDMP IDOF: character(4): Degree of freedom IDOF = IRX: Rotation around local x-axis IDOF = IRY: Rotation around local y-axis IDOF = IRZ: Rotation around local z-axis IDOF = IRYZ: Rotation around bending axis IBOUND: integer: Constraint IBOUND = -1: Fixed (Legal if 2 of 3 dofs are fixed) IBOUND = 0: Free. Not available with IDOF = IRYZ IBOUND = 1: Constant stiffness IBOUND > 1: Table with IBOUND pairs of moment - rotational angle to be specified RAYDMP: real: Stiffness proportional damping coefficient 3 or 2 input lines to be specified: IRX, IRY, IRZ or IRX, IRYZ x, y and z-axes refer to the local coordinate system of the element to which the flex joint is attached. This is similar to the ball joint connector as illustrated in the figure Rotation freedom for a ball joint component. 5.6. Stiffness data Stiffness data are to be given in the sequence IRX, IRY and IRZ or IRX and IRYZ. Stiffness data are to be omitted for IBOUND ⇐ 0 Linear stiffness IBOUND = 1, One input line ~ STIFF ~ STIFF: real: stiffness with respect to rotation \(\mathrm {[FL/deg]}\) Nonlinear stiffness; IBOUND > 1 IBOUND > 1, IBOUND input lines ~ MOMENT ANGLE ~ MOMENT: real: Moment corresponding to rotational angle \(\mathrm {[FL]}\) ANGLE: real: Rotational angle \(\mathrm {[deg]}\) MOMENT and ANGLE must be given in increasing order. Linear extrapolation will be used outside the specified range of values. For dofs IRX, IRY and IRZ, both negative and positive values should be given. For dof IRYZ, MOMENT and ANGLE have to be greater or equal to zero. To avoid convergence problems, the first pair should be 0.0, 0.0. 6. FLUID - Specification of internal fluid flow 6.1. Data group identifier NEW COMPonent FLUId 6.2. Component type number CMPTYP-ID CMPTYP-ID: character(8): Component type identifier 6.3. Fluid flow characteristics RHOI VVELI PRESSI DPRESS IDIR RHOI: real: Density \([\mathrm {M/L^3}]\) VVELI: real: Volume velocity \([\mathrm {L^3/T}]\) PRESSI: real: Pressure at fluid inlet end \([\mathrm {F/L^2}]\) DPRESS: real: Pressure drop \([\mathrm {F/L^3}]\) IDIR: integer, default: 1: Flow direction code 1: Inlet at supernode end 1 of the line 2: Inlet at supernode end 2 of the line The pressure drop is assumed to be uniform over the line length. For further clarification of pressure definition, confer Theory Manual. In this version only RHOI is used to calculate weight and mass for static and dynamic analysis. The other parameters are used for calculating wall force (flange force) only depending on output option (OUTMOD) 7. EXT1 - External wrapping This component can be used to model additional weight or buoyancy modules attached to a riser line. The specified additional weight and buoyancy are used to adjust the average properties of the segment. 7.1. Data group identifier NEW COMPonent EXT1 7.2. Component type identifier CMPTYP-ID CMPTYP-ID: character(8): Component type identifier 7.3. Mass and volume AMS AE RGYR FRAC AMS: real: Mass per unit length \(\mathrm {[M/L]}\) AE: real: Buoyancy volume/length \(\mathrm {[L^2]}\) RGYR: real: Radius of gyration around local x-axis \(\mathrm {[L]}\) FRAC: real: Fraction of the segment that is covered \(\mathrm {[1]}\) 0 ⇐ FRAC ⇐ 1 The resulting properties of the segment with external wrapping are: Mass / length: \(\mathrm {AMS_{res}=AMS_{cs}+AMS_{ext}\times FRAC}\) Resulting radius of gyration: \(\mathrm {RGYR_{res}=(AMS_{cs}\times RGYR_{cs}^2+AMS_{ext}\times RGYR_{ext}^2\times FRAC)/(AMS_{cs}+AMS_{ext}\times FRAC)}\) Resulting external area for buoyancy: \(\mathrm {AE_{res}=AE_{cs}+AE_{ext}\times FRAC}\) Where: cs denotes the original cross section properties; i.e. without wrapping. ext denotes the properties of the wrapping given in this data group. res denotes the resulting average segment properties Figure 6. Description of external wrapping 7.4. Hydrodynamic coefficients CDX CDY AMX AMY CDLX CDLY CDX: real: Drag force coefficient in tangential direction \(\mathrm {[F/((L/T)^2\times L)]}\) CDY: real: Drag force coefficient in normal direction \(\mathrm {[F/((L/T)^2\times L)]}\) AMX: real: Added mass per length in tangential direction \(\mathrm {[M/L]}\) AMY: real: Added mass per length in normal direction \(\mathrm {[M/L]}\) CDLX: real, default: 0: Linear drag force coefficients in tangential direction \(\mathrm {[F/((L/T)\times L)]}\) CDLY: real, default: 0: Linear drag force coefficients in normal direction \(\mathrm {[F/((L/T)\times L)]}\) The coefficients specified for the external wrapping are added directly to the coefficients specified for the pipe. The drag forces per unit length acting in the local x-direction is computed as: \(\mathrm {F_x=(CDX_R+FRAC\times CDX)VRELX\times VRELX+(CDLX_R+FRAC\times CDLX)VRELX}\) In the case of an axis symmetric cross section the drag force per unit length, \(\mathrm {F_n}\), acting normal to the local x-axis is computed by assuming that the instantaneous drag force direction is parallel to the instantaneous transverse relative velocity component \(\begin{array}{l}\mathrm {F_n=(CDY_R+FRAC\times CDY)(VRELY^2+VRELZ^2)}\\+\mathrm {(CDLY_R+FRAC\times CDLY)\sqrt{VRELY^2+VREZ^2}}\end{array}\) In the case of a cross section with 2 symmetry planes the drag force per unit length in the local y and z directions are computed as: \(\mathrm {F_y=(CDY_R+FRAC\times CDY)VRELY\times VRELY+(CDLY_R+FRAC\times CDLY)VRELY}\) \(\mathrm {F_z=(CDZ_R+FRAC\times CDZ)VRELZ\times VRELZ+(CDLZ_R+FRAC\times CDLZ)VRELZ}\) Where: \(\mathrm {CDX_R,CDY_R,CDZ_R}\): are the input quadratic drag force coefficients of the riser in local x,y and z-directions \(\mathrm {CDLX_R,CDLY_R,CDLZ_R}\): are the input linear drag force coefficient of the riser in local x,y and z-directions \(\mathrm {CDX,CDY}\): are the input quadratic drag force coefficients of the external wrapping in local x and y-directions \(\mathrm {CDLX,CDLY}\): are the input linear drag force coefficients of the external wrapping in local x andy-directions \(\mathrm {VRELX,VRELY,VRELZ}\): are relative water velocities in local x, y and z-directions For an axis symmetric pipe with external wrapping the input drag force coefficient in normal direction can be calculated as: \(\mathrm {CDY=\frac{1}{2}\rho (DC_{dn}-D_RC_{dnR})}\) The added mass per unit length in normal direction can be calculated as: \(\mathrm {AMY=\rho \frac{\pi }{4}(D^2C_{mn}-D_R^2C_{mnR})}\) where: \(\mathrm {\rho }\): water density \(\mathrm {D}\): outer diameter of the external wrapping \(\mathrm {D_R}\): outer diameter of the pipe \(\mathrm {C_{dn}}\): normal drag coefficient \(\mathrm {C_{dnR}}\): normal drag coefficient of the pipe \(\mathrm {C_{mn}}\): normal added mass coefficient of the external wrapping \(\mathrm {C_{mnR}}\): normal added mass coefficient of the pipe