f 1111111i~~ua::Ww1 ~q.tj. m1 9 m1l.m 1 imu1w 2529 2 STATC EQULBRUM OF MOORNG LNES BY FNTE ELEMENTS by Somchai Chucheepsakul Lecturer, Dept. of Civil Engineering King Mongkut 's nstitute of Technology Thonburi Bangkok 10140, Thailand Abstract This paper presents a method of static analysis to determine the equilibrium position of mooring lines. n the analysis, a functional is introduced to suit the problem physically, and stationary condition of the functional and equilibrium equations are used to solve the problem. Application of the finite element method yields a system of nonlinear equations and the Newton-Raphson iterative procedure is used to solve them. Accuracy of the model is verified by comparison of the numerical results with the catenary and neutrally-buoyant cable.
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3 ntroduction Mooring lines or cables are tht;: fundamental structures used in many offshore engineering problems. Although these structures are restricted to a simple configuration, the analytical solutions are complicated due to the applied loads. Explicit solutions can be obtained only in the simple cases. For general solutions, numerical techniques must be employed. Literature on mechanics of mooring lines can be found in the text book by Berteaux [ l ]. n the static analysis, there are three loading cases to be considered. First the cable is subjected to its own weight which yields the catenary solution. Secondly, only horizontal force due to the steady current is considered which is the case of the neutrally buoyant cable. The combination of first and second loading case yields the general condition of the cable in which the solutions can be obtained through the numerical techniques. Pode ( 5] and Wilson [7] established the tables of integration resulting from the numerical evaluation, known as cable functions for the cases of constant and variable tangential drag respectively. Although those tables have been used in the design of mooring lines for many years, they are limited to the uniform current profile and not convenient for repetitive design works. Seek-Hong [6] proposed a rigorous analysis which is more general than that given in text book but there is no numerical example demonstrated. The method of analysis presented here involves the solution of the stationary condition of a functional and equilibrium equations. n the analysis the vertical distance from the seabed is used as the independent variable instead of arc length. Thus, at a given elevation the horizontal coordinate defines the equilibrium position. The finite element method is used to solve the stationary
4 condition. The resulting algebraic equations are nonlinear and the Newton- Raphson iterative procedure is used to solve them. Equations of Equilibrium The simple buoy system considered here is shown in Fig. 1. The bottom end is the anchor point while the top end is connected to the buoy where the known tension TH is applied to maintain the equilibrium at the specified position. The horizontal component of the top tension considered as the drag force acting on the buoy can be determined if the inclination angle at the top is given. The load intensities per unit arc length in the horizontal and vertical direction are q and w respectively. The equilibrium configuration of the cable is expressed in rectangular coordinates. t is intended to express the horizontal coordinate x in term of the vertical coordinate z. This means that every quantity will be parameterized in z, instead of the arc length s. z Tension x(z) ' H z x Fig. 1 Cable displacement configuration
5 in Fig. 2. Consider an infinitesimal element at the equilibrium position as shown dx T+dT. --ids T?;..~ Tstne Fig. 2 Cable differential element Equilibrium of forces in vertical and horizontal directions yield ~ (T sine) w (1) and d as (T cos 0) -q (2) in which s represented the stretched arc length. These two equations can not be solve explicitly since the total arc length considered in the problem is unknown. Solution methods given in Refs. [1] and [6] may be applied to solve the problem numerically. Functional 11 For more convenient, an alternative method of solution is introduced m which the following functional is used :
l 1 1 i t 1 6 L 11 t (T -qx) ds (3) in which L is the total arc length within the depth H, T is the tension at any position and ds is an infinitesimal arc length in an infinitesimal depth dz which can be expressed as..1 +x' 2 dz. Functional u used in this analysis is modified from Refs. [2] and [3], by neglecting those bending terms. By changing of the independent variable from s to z, the upper integration limit in Eq. (3) is changed from L to H. U Therefore, Eq. (3) becomes H f (T-qx) j 1 + x ' 2 dz 0 (4) and the total cable length can be determined fro.m L H 0 in which x' denotes #z f j 1 + x' 2 dz (S) Cable Tension The cable tension at z. can be obtained from the equijibrium equations as follows. Let Th and T v be the components of T in the horizontal and vertical directions respectively, in which Th T cose and T v T sire. Thus, T Tv sine + Th cose (6) Differentiate Eq. (6) with respect to s, utilizing Eqs. (1) and (2), and noting dz dx that sin e as and cos e as ' one has
dt ds dz dx w- -qds ds 7 (7) 1 ' f w and q are constant, integrate equation (7) from the location z to the top, the tension T can be written as T(z) T 1-1 + w (z - H) - q [x (z) - x (H)] (8a) in which TH is the tension at the top. f w and q are not constant, then T(z) TH- JH wet.;+ H q(cn) d. df; (Sb) z z in which E; S a dummy variable to replace z. Therefore, if the displacement x is know~, the tension T can be determined by Eq. (Sa) or (Sb). Solution Technique The finite element discretization for x(z) is used to solve the condition on 0. The coordinate x(z) is composed of two parts. x (z) (9) The component x fl is linear in z while xa is nonlinear. The entire region, 0 < z < H, is divided into N subregions, or elements. Thus, the functional ff is approximated as 1l (10) in which ff k 1s the part of the functional associated with the kth element. A typical element is shown in Fig. 3. The coordinate component xa is approximated by a cubic polynomial in the dimensionless parameter r;. Thus, the component x a in the element is approximated by x l N J { d} a ( 11)
8 in which L N J is the interpolation- matrix and {d} is the local degrees of freedom written as { d} [ x.(o), x (1), a dxa (l)j dz. T ( 12) z r,! x,' a, -. ----~-!--, x ~ig. 3 Typical element and coordinates From Eq. ( 3), the contribution to the global equilibrium equations from the kth element is { auk} ad; which can be expressed as ( 13)
[KNL] " [:'~ ~dj l ( [l N' f ( +x'\32 LN' J]. hd~ (17) 9 m which x' x; + x~. Note that x; is constant while x; is l N' J { d}. t is notice here that q J 1 + x' 2 is not differentiated, for details explanation see Refs. [ 3] & [ 4 ]. The global equilibrium condition 611 0 can be written in terms of global coordinates { D} { ~"o.} {O} l (14) The boundary conditions are as follows x (0) a x (H) a 0 ( 15) Theyarederived from the condition x(o) 0 and x(h) xh. ncremental Equation The set of equations (14) is nonlinear; therefore the Newton- - Raphson iterative procedure is used. The incremental equation used in the procedure is -{ R} ( 16) The square matrix [ KNL] corresponding to the kth element is and the column vector {R} is shown in Eq. (13). Equations (13) and (17) can be numerically evaluated by Gaussian quadrature integration. The solution step of Eq. ( 16) can be found in Ref. [2 ]. The results give dx the components x and ~ at the node points,.and the nodal coordinates a dz x is equal to the sum of xfl and xa.
10 Nl!merical Examples To verify the model formulation, two loading cases for which explicit solutions obtained from Ref. [6] were compared. First the cable is suspended by its own weight, the solution yields the catenary. By neglecting q in Eqs. (4) and (8), the numerical results obtained from the computer program and the catenary equations are compared and listed in Table 1. n the computation, the number of equal length finite elements is 20 and the number of iterations is 6. t is obvious that the results are very close to the exact ones, and only small discrepancy occurs near the touch down point in which dx dz to infinity. The configuration of a catenary is shown in Fig. 4. Table 1. Comparison of numerical results for a catenary. ~ is close Node z(m) x(m) Numerical Ref. [6] 21 40.00 20.63437 20.63437,, 17 34.00 18.93213 18.94559 13~-- 2J:LOO 16.86194 16.89236 9, 22.00 14.20040 14.25417 5 16.00 10.37601 10.46968 1 10.00' 0.00000 0.00000 Cable Length (m) 38.6385 38. 7298
z 11 20.63437 m. 40 E N Cll - u c,,, µ ll... Q 30 20 T 0 To x v- cosh (~} To -1 To z - cosh (~} w T To 0 10 s. - sinh (~) w To To 0.5 kn w 0.05 knm 0 1 0 20 30 x Displacement, x(m) Fig. 4 Catenary cable Second case, the cable is equilibrium in vertical direction which means the sum of buoyancy force and cable weight is zero. The cable is said to be neutrally buoyant. n the formulation w is zero but q is not necessary zero. Figure 5 shows the configuration of this type of cable for various values of q and the numerical results for q equals to 0.001 knm are compared and listed in Table 2. A good a~reement was obtained with 20 elements and 6 iterations
12 Table 2. Comparison of numerical results for a neutrally-buoyant cable. - Node z(m) x(m) Numerical Ref. [ 6] 21 600.00 503.0476 503.0476 17 480.00 441.9376 441.9320 13 360.00 362.6107 362.6046 9 240.00 263.7049 263.6987 5 120.00 143.5225 143.5162 1 0.00 0.0000 0.0000 Cable Length (m). 791.2645 791.3740 z 503.0476 600 500 ~ E N ~ tlj u c J.J "' Ol.-! Q 400 300 q q 0.003 knm 200 '.r.v x -'--- (cosec e - cosec 0) q 0 Tv 0 Bo z ~ (ln(tan -)- ln(tan~)] ft 2. 2 s...::!.. (cot 0 - cot 0) q 0 100 TH 1.0 kn 200 300 400 500 600 x Oisplacament, x(m)
13 z 503.0476 m 600 500 s -N. Q u c:..., ' Ul... Q 400 300 TH 45.0 kn 200 "' 0.05 knm q 0.001 knm 1 00.200 300 400 500 600 x. Displacement, x(m) Fig. 6 General equilibrium configuration When the cable is subjected to the loading in. both directions, this is the general condition. The cable can be in a slack-moored or taut - moored shape depended on the top tension. Figure 6 shows the equilibrium configuration for a cable with TH 45 kn, w 0.05 knm, q 0.001 knm, and xh 503.0476 m. Conclusions The method of analysis presented here is an alternative one to determine the equilibrium configuration of mooring lines o r cables. n the formulation, a suitable functional coupled with equilibrium equations are used. The finite element method and. Newton- Raphson procedure are used to solve the problem numerically. Numerical examples are
14 Appendix.- References.. 1. Berteaux, H.O., "Buoy Engineering, " John Wiley & Sons, New York, N.Y. 1976, pp. 97-134. - 2. Huang, T., and Chucheepsakul, S., "Large Displacement Analysis of a Marine Riser," ASME journal of Energy Resources Technology, Vol. 107, Mar. 1985, pp. 54-59. 3. Huang, T.," On Large Displacenent Analysis of a Class of Beams," ASCE Proceedings of the Fifth Engineering t~echanics Division Specialty Conference, Aug. 1984, pp. 248-251. 11 4. Huang, T., On the Functional m a Marine Riser Analysis,". Unpublished paper,. The University of Texas at Arlington, Arlington, Tex., 1985. 5. Pode, L., "Tables for Computing the Equilibrium Configuration of a Flexible Cable in a Uniform St ream, " Report 637, David Taylor Model Basin, Dept. of the Navy, Washington, D.C., Mar., 1951. 6. Seek-Hong, C., 11 Mechanics of Statics Catenary with Current Loading,". ASCE journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 109, No.3, Aug., 1983, pp.340-349. 7. Wilson, B. W., "Characteristics of Anchor Cables in Uniform Ocean Currents, 11 Te.chnical Report No. 204-1., Texas A & M Research Foundation, College Station, Tex., Apr., 1961.
l Appendix. - Notation 15 The following symbols are used in this paper; { D} { d } H h [KNL] L N [N] q { R} s T T H T 0 w x 6 e f; r; 11 ::: global degrees of freedom; local degrees of freedom; water depth; element length; nonlinear stiffness matrix; cable length; number of subdivision; interpolation matrix; horizontal load intensity per unit arc length; column vector; arc. length; cable tension; cable top tension; cable bottom tension; vertical load intensity per unit arc length; horizontal coordinate; buoy excursion at the top; vertical coordinate; variation symbol; inclination angle; dummy variable to replace z; dinensionless parameter zh; and functional.