Computers & S/rurrurrs Vol. 30. No. 5. pp. 1079-1089. 1988 Printed in Great Britain.

EVALUATION OF FORCE-DISPLACEMENT

0045-7949/88 S3.00 + 0.00 0 1988 Pergamon Press plc

RELATIONSHIP

FOR MULTICOMPONENT MOORING CABLE BY F.E.M.

R. P. SINGH and J. VERMA

Department of Civil Engineering, I.I.T. Delhi, New Delhi 110016, India

(Received 6 November 1987)

Ahstracct-The integrity of moored offshore platforms held over subsea oil well heads is largely dependent on the characteristics of the mooring system. In the present work a deep sea multicomponent mooring line is analyzed by F.E.M., taking both the straight line elements and curved element into consideration. The force-displacement relationship is obtained for the guy attachment point with the supported structure and a set of results are obtained to study the effects of various parameters of the mooring line on its static response.

I. INTRODUCHON

The offshore philosophy of fixed base structures normally employed for oil and gas production in shallow waters has changed with the increase of industrial activities in deeper waters. New types of platforms such as the guyed tower/floating platforms are held over subsea well heads by a system of mooring cables. An appropriate knowledge of the mooring cable characteristics is of vital importance in understanding the overall behaviour of the sup- ported structure subjected to ocean environmental/ operational loads. The motion of a platform is di- rectly dependent on the capacity of ‘the mooring system to resist its movements caused by various loads. In the present work the force-excursion relation for a deep sea mooring line (Fig. 1) at its suspension point with the platform is obtained by using the finite element method [l], which can account for the nonlinearities arising from the geometry, the material, the boundary conditions and the fluid loadings. Both the straight line elements and the curved element are employed in obtaining the results. The present solution procedure is based on the stable static configuration of the mooring line in its neutral position, which is efficiently obtained by employing the new technique [2] developed by the present authors. A parametric study is carried out in order to estimate the effects of different lengths, submerged unit weight and axial rigidity of the various components of the mooring line on its force- displacement relationship.

2. DISCRETIZATION OF MOORING CABLE

The mooring line shown in Fig. 1 has three com- ponents: (i) the anchor cable, normally resting on the sea bed; (ii) the clump weights partially lying on the sea bed and partially hanging in the water; and (iii) the guy cable hanging in the water from the sus- pension point of a platform. Under extreme ocean environmental conditions the anchor cable and/or clump weights may get fully lifted in the water except

at the fixed anchor point. When the anchor cable is lying on the sea bed it is taken as a straight line element, otherwise as a curved element. The clump weights are treated as either a typical element (Fig. 2c) which is partially straight and partially curved or a fully curved element (Fig. 2b) when it is completely lifted from the sea bed. The guy cable is discretized into a number of straight line elements which enables the simple calculation of hydro- dynamic forces on the cable due to ocean waves and currents. The fluid forces on anchor cable and clump weight may be neglected under deeper water con- ditions. The frictional force between a cable and the ocean floor is not taken into account.

3. FORMULATIONS OF STRAIGHT LINE ELEMENT

Webster [l] treated in detail the subject of finite element method applied to cable problems. As per his derivations the tangent stiffness matrix of a one dimensional element in deformed configuration C, (Fig. 2a) with material reference C, consists of four components

VGI = [&I + Kl+ Kl+ &I. (1)

The small displacement matrix can be written as

r I 0 -I 01

(14

The large displacement matrices can be written as

[K,] q f 0 2 1 0

r 28, e2 -28, -e,l

X e2 0 -e2 0

-28, -e2 28, e2 (lb)

1 -e2 0 0, 0J c A.S 30 SE 1079

1080 R. P. SINGH and J. VERMA

Fig. 1. Schematic view of mooring system.

- [K2] =y ; 0 =

t II

I e; e,e* 4: -e,e*

X e,e2 e: -e,e2 -e: -0: -e,e2 e: e,e2 I (14

-e,e2 -e: e,e2 e:

8 - Ul 8, =- 1,

and

v2 -4 e2=1. r

(14

The higher order matrices [K,] and [K2] become null matrices in case the deformed configuration C, and material reference configuration C, coincide with each other

The initial stress matrix

r ’ o-1 01

0 -1 i&l=; _; ; 1 o . (le)

,

1 O-l 0 11

The total axial force,

- Pi = EActola,. (2)

The total Green’s strain,

(3)

configuration C, with material reference C, is given by

(4)

The deformed length of the element

f, = [(f, + z42 - u,)2 + (v2 - v$]“*. (5)

In the above equations, l,, I,, 1, = natural, reference configuration and deformed configuration length of - the element; EA = axial rigidity and u,, u,, v,, v2 = nodal displacements as shown in Fig. 2a.

4. FORMULATIONS OF CURVED ELEMENT

The governing formulations presented here were employed in the solution of cable problems by O’Brien and Francis [3], O’Brien [4], Jennings [5] and Jayaraman [6]. In the case where the elastic catenary effect is small the equations for extensible cable geometry (Fig. 2b) can be written as

sinh2 1 I,= V2fH2---Z--

F,=;(Vcoth1. +I,)

F4=wl,,-F2; F,= -F,

Tf=F:+F:; T;=Ff+F;

The internal force vector for an element in deformed

Tj - T; T, - TI v=m+-

W

Force-displacement relationship for multicomponent mooring cable 1081

AB :Ir, Length tn reterence conflgwation , CC,)

A’ B’ = 1, , Length in deformed configuration , (Ci)

f B’ ; I,.,, Length I” deformed conf lguratm ,(Ckl,1)

Fig. 2a. Straight line element.

F2

t

+ ‘; “0

‘/ / Fj ----tX

*

Fig. 2b. Curved element.

Fig. 2c. Combined straight line and curved element.

H=F, &+;,og- [

T, + F, T, at node J; F,, F4 = horizonal and vertical com-

T, - Fd 1 ponents of cable-force T, at node I; and m = axial rigidity.

I, = I, + & [

G + FZ F,T, + F,T, + F; log T-F . 1 (6)

The tangent stiffness matrix of a curved element in

I 4 deformed configuration (Fig. 2b) can be written as

In the above equations, for a cable: w = the weight per unit length; 1, = the natural length; f, = extended length; H, V = horizonal and vertical distance be- Kl = tween the suspended nodes I and J (Fig. 2b), F,, F, = horizonal and vertical components of cable-force

1082 R. P. SINGH and J. VERMA

where

Hk Fig. 3. Static configuration for different position of suspension point.

The change in the nodal forces F, and F2 caused by the differential change in horizontal and vertical distances (H, V) between the nodes J and I is calcu- lated from

u2=%= -52155

and (*)

I

5. FORMULATIONS OF COMBINED STRAKXT LINE-CURVED ELEMENT

The combined straight line-curved element ‘BCD’ is shown in Fig. 2c. Whereas the portion ‘BC’ of the element is straight the other curved portion ‘CD’ has _

55 =5,54-52b.

zero slope at point C. The general equations combined geometry are written below.

600

500 -

.

Loo--

;

r 3oo $ -

for the

-@&I I I 1 I I I I - 10 -0 -10 -20

Vertical excursion t Ml

Fig. 4. The relationship between different forces and vertical excursion of a single mooring line at guy attachment point.

Force-displacement relationship for multicomponent mooring cable 1083

The natural length of portion ‘CD’ = F,/W where &, E = natural length and axial rigidity of the anchor cable.

(9)

where: w = weight per unit length; 7, = the natural length of ‘BCD’; j?, P = horizonal and vertical dis- tance between the end nodes B and D respectively; F,, F2 = horizontal and vertical components of cable- force, TD at node D; F, = cable-force at node B and - EA = axial rigidity of the element. The tangent stiffness matrix of a combined straight line-curved element can be written as

I&] _I -1 _; Z],

where

and

(10)

In the case where the anchor cable ‘AB’ is also taken into consideration the expressions for &, e, and & remain the same as those of [,, f, and [, shown in eqns (11) and the relations for R and [, become

(11)

6. ITERATIVE SOLUTION PROCEDURE

For a known deformed configuration of a mooring line the tangent stiffness matrices, the internal force vectors and the external force vectors are generated in material reference coordinates and transformed to global coordinates, if necessary for all the straight line elements, and combined with those of curved elements in order to obtain the overall structural properties of the mooring line (Fig. 1). The self- weight of a straight line element is equally lumped at its two ends, to generate the associated external load vector, and in the case of curved elements the external force vector is a null vector. The discrete form of equilibrium equations [I] to be solved for the known configuration Ci of the mooring line during the iterative solution process can be written as

:+ ‘(li} = I{ti} + ;+ ‘{Li}, (13)

where i[R,] = the tangent stiffness matrix for the structural configuration C, with material reference configuration C,; f{G} = the nodal displacement vector measured from the material reference con- figuration C, to that corresponding to the deformed configuration Ci; i{f,} = the total external force vec- tor corresponding to the deformed configuration C,; and :{A} = the internal load vector corresponding to the deformed configuration C,. Equation (12) is solved iteratively until the residual force vector shown on its right hand side and the residual dis- placement vector :+ ‘{G} become negligibly small.

7. PRESENT SOLUTION APPROACH

The static stable configuration of the mooring line (Fig. 3) in its neutral position and subjected to self weight is obtained in order to start the evaluation of force-excursion relation at its attachment point E. When the actual static shape C, is known the internal and external nodal forces including those of support reactions balance each other and as such the residual force vector becomes negligibly small. In this case eqn (12) can be written as

where {SC,} = change in the displacement vector for all the nodes except those for the attachment node E; {&i2} = change in the displacement vector for the attachment node E; and [R,,], [Ki2], [KZ,], [K22] = the corresponding submatrices of the tangent stiffness matrix [R] of the mooring line.

1084 R. P. SINGH and J. VERMA

From eqns (14) one can write 8.1 General force-excursion characteristics

If the displacement vector {6tir} for the attachment node E is specified, the displacement vector (Si&} can be calculated. The addition of the vector {G,, c&>’ with the known nodal coordinates of the static shape provides the next non~uiIib~~ initial configur- ation of the mooring line with the displaced position of the attachment point E. The iterative solution procedure is applied until the residual force vector and differential displacement vector become neg- ligibly small. The whole solution process is repeated for each applied small displacement of attachment node E. As the perturbation given to the structural system is not taken to be large the convergence of the solution process is assured.

8. RESULTS AND DISCUSSIONS

The guy cable of the mooring line ABCDE (Fig. 1) analysed is divided into 10 straight line elements. The height from the sea bed to the still water level is taken to be equal to 440 m and the horizontal distance between the anchor point A and the attachment point E is equal to 2200m. In general, except where otherwise stated, the natural length of the anchor cable, clump weights and guy cable is taken in turn equal to 800 m, 40 m and 1420m. The submerged - unit weight ‘w’ and the axial rigidity ‘EA’ are kept the same for both the anchor and guy cables and these parameters are equal to 0.0293 t/m and 67,200 t respectively. The clump weight is assumed to be inextensible and its submerged unit weight equals 2.5 t/m.

The suspension point E is ailowed to move separ- ately in the horizontal and vertical directions; the movement in each step and the range of excursion are taken in turn equal to 1 m and 20 m. The cable force at the suspension point E and its horizontal and vertical components versus the horizontal excursions curves are shown in Fig. 5; the corresponding curves for the vertical displacements of the attachment point are presented in Fig. 4. In general the force- displacement relationship is nonlinear in nature. The nonlinear effect is large in the case of horizontal excursion of the attachment point; the influence decreases significantly after the complete lifting of the clump weights from the sea bed. However, the effect in the case of the vertical force component is smaller than that of the horizontal component of the cable force. As the slope of the force-excursion curve is the largest around the zero displacement position of the attachment point, the mooring line’s resistance to the movement of the suspension point from neutral position is maximum and the mooring system be- haves in stiff mode. The resistance decreases dras- tically after the complete lifting of the clump weights, causing softening of the mooring line. All the force-excursion relationships (Fig. 4) corresponding to the vertical displacements of the attachment point are almost linear. The influence of horizontal excur- sion on complete liftin of the clump weights is greater than that of the vertical excursion of the suspension point. Whereas the clump weights lifting phenomenon occurs for a vertical displacement of around 12 m, the corresponding figure is only 4 m in the case of horizontal displacement of guy attach- ment point.

Description of torep 1 Legend l

Horizontal t-

Vertical _____

Resultant -.-

Complete Wing of clump weight

Complete lifting of anchor cable

L I t

920 I I t I 1

-10 0 10 20

Horitontd excursion IT4

Fig. 5. The relationship between different forces and horizontal excursion of a single mooring line at guy attachment point.

Force-displacement relationship for multicomponent mooring cable 1085

0 I I # 1 I 1 1

-25 -10 0 15 25

Horizontal excursion (t+O

Fig, 6. Inlluence of guy c&e axial rigidity(m) on the relationship between diEerent forces and horizontat excursion of a single mooring line at guy attachment.

8.2 I$uence of cabie axial rigidiiy To study the influence of axial rigidity ‘m’ on the

structural response of a mooring line, the results are obtained with three different values of ‘m for the guy cable and the anchor cable separately. The basic value of the axial rigidity *EA ’ is equal to 57,200 t; the other two values are taken equal to two and three times the basic value of n. The other properties of the structural remain unchanged. The nodal forces versus horizontal excursion curves associated with the three cases of guy cable axial rigidity are shown in Fig. 6; the co~es~uding curves in the case of vertical excursion are presented in Fig. 7. In general the nodal forces at the suspension point increase with

the increase in the value of ‘m’; when the excursion is large in the negative direction the effect is observed to be quite small. The increase in the cable forces in the case of ‘2m’ over that of case ‘Eit’ is much larger than the corresponding increase in case of ‘3m’ over that of case ‘2m’. For the zero displace- ment positions of the guy attachment point the horizontal component of the cable force in case of ‘2m’ is 25.7% greater than that of ‘m’; the corre- sponding figure in the case of ‘3m’ over the case of ‘2m’ is only 5.4%. These figures for the vertical force ~m~onent are 21.6 and 4.3% respectively. This implies that if the actual value of ‘m’ is greater than a certain limiting value of axial rigidity then the cable

t 40

I I 1

-10 5 IO 20 Verticd excursion (MI

Kg. 7. Influence of guy cable axial rigidity (B) on the relationship between different forces and vertical excursion of a single mooring tine at guy attachment.

1086 R. P. SINGH and J. VERMA

0 Complete lifting of clump weight

A Complete lifting of anchor cable

-20 -10 0 10 20 Horizontal excuwon 1 M 1

Fig. 8. Influence of anchor cable axial rigidity (E-f) on the relationship between different forces and horizontal excursion of a single mooring line at guy attachment.

may be treated as inextensible in the analysis of the mooring line. The stiffness of the mooring system also increases with the increase in the axial rigidity and the effect is maximum for the excursion around the zero displa~ment point but before the complete lifting of clump weights. The lifting phenomenon of clump weights is greatly influenced by the axial rigidity of the guy cable and it occurs in cases of ‘EA’ ‘2m and ‘3m’ for horizontal excursions in turn equal to + 3.9, -0.2 and - 1.8 m; the corresponding figures for the vertical excursion are + 11.2, - 1.4 and - 5.7 m respectively. The force-excursion curves for the different axial rigidities of the anchor cable are presented in Figs 8 and 9. Although the behaviour of the mooring line due to the change in the axial

rigidity of anchor cable is almost similar to that of the guy cable discussed above, the effect of ‘m is smaller in this case. For the zero displacement posi- tion of the guy attachment point the horizontal component of the cable force in the case of ‘2m’ is 13.1% greater than that of ‘EA’; the corresponding figure, for the case of ‘3j-Z’ over that of the case ‘2m’ is 5.3%. In the case of the vertical excursion of the suspension point these increases are 10.7 and 4.3% respectively.

8.3 rn~uence ~~~1~~ weighty-submerged unit weight

Results are obtained for the mooring line with three different submerged unit weights (1.5, 2.0 and 2.5 t/m) of clump weights; the other structural prop-

Description d Lqlond for ‘m’ IT) force 67200 I13Lf.00 i 2OlWO

Horizontal - _____ -*_

Vorticol - I --_-._ -.-

c A 500 ,

Complete lifting of clump weight

Complete lifting of anchor cable I

t -020

I I I 1 I I I I - 10 0 10 20

Vertical excursion tM1

Fig. 9. Influence of anchor cable axial rigidity (m) on the ~lationship between different forces and vertical excursion of a single mooring line at guy attachment.

Force-displacement relationship for multicomponent mooring cable 1087

Complete lifting Of clump weight

a0 A Complete lifting of anchw cobte

5

jj 300

t

1 -020

I I I I I I -10 0 10 20

Horizontal excursion ltd)

Fig. 10. Influence of clump weight (w/l) on the relationship between different forces and horizontal excursion of a single mooring line at guy attachment point.

erties of the system are kept the same. The nodal forces versus horizontal excursion curves for all cases of unit weights are presented in Fig. 10; the corre- sponding figure for vertical excursion is shown in Fig. II. In the positive excursion range the cable force increases with the increase in the unit weight of the clump weights; these curves for different unit weights are almost identical in the negative excursion range. This happens because the clump weights inertia force becomes ineffective in contributing to the guy cable force. The excursion of suspension point E corre- sponding to complete lifting of clump weights from the sea bed increases with the increase in the unit weight and this displacement varies almost linearly with the unit weight of the clump weights. This results

in the increase in the stiffness range of the mooring line around its zero displacement position. At the complete lifting point of the clump weights the horizontal excursions in the case of unit weights of 1.5,2.0 and 2.5 t/m are 1 .O, 2.0 and 4.0 m res~ctively; the corresponding values for horizontal components of guy cable force are in turn equal to 276.9, 323.4 and 378.2 t. The vertical excursions in the case of these three different clump weights are in turn 1 .O, 5.0 and 11Sm at the lifting point. In these cases the vertical component of guy cable force is always equal to the total weight of the clump and guy cable together. Even though, the stiffness of the mooring system is almost the same up to the lifting point, the resistance to movements beyond lifting of the clump

m f

Description LPw%d for *W’(T/M) of force 1.50 I 2.00 I 2.50

Horizontal _.- ___- -

Vertical -.- 1 -____ - -0 COmplete liftifI$I of clump weight

I UM-

COmplete lifting of anchor cable

91 I I I I I I

” 0 -10 0 10 20

Veriicot excursion I Mf Fig. 11. Influence of clump weight (w/l) on the ~lationship between different forces and vertical excursion

of a single mooring line at guy attachment point.

R. P. SINGH and J. VERMA

Complete lifting of clump might

Complete lifting of anchor cable

I -020

I I I I I I 1

-10 0 10

Horizontal excursion It.41

Fig. 12. Influence of anchor length (6) on the relationship between different forces and horizontal excursion of a single mooring line at guy attachment point.

weights increases quite significantly with the increase in the total weight of the clump weights.

8.4 Influence of anchor cable length

To examine the effects of anchor cable length on static response of mooring line three different lengths of 600, 800 and 1000 m are taken into consideration in obtaining the results; the position of the node ‘B’ (Fig. 1) joining the anchor cable and the clump weights is kept unchanged. The cable forces versus horizontal excursion curves are presented in Fig. 12. The effect of anchor cable length on the horizontal component of guy cable force is important; the influence on its vertical component is found to be

small. Although the larger stiffness-range around the neutral position of the mooring system increases with the increase in the length of anchor cable, the stiffness itself is subjected to a diminishing effect. At the lifting point of the clump weights the horizontal excursions are 2.9,4.0 and 5.3 m respectively for cable lengths of 600, 800 and 1000 m. In these length cases and zero displacement position of point E the horizontal com- ponent of the guy cable force is in turn equal to 307.0, 287.1 and 270.2 t; the corresponding figures for its vertical component are 119.6, 113.3 and 108.0 t, respectively. The characteristics of cable force versus vertical excursion relation presented in Fig. 13 are almost similar to those discussed for the case of

500 - _____ -.-

Vertical - _____ -.-

Complete lifting of clump weight

400 Complete liftmg of anchor cable

I -020

I I I I I I

-10 0 10 20

Vertical excursion I Ml

Fig. 13. Influence of anchor length (6) on the relationship between different forces and vertical excursion of a single mooring line at guy attachment point.

Force-displacement relationship for multicomponent mooring cable 1089

horizontal excursion. The linear characteristic of these curves remains unchanged with the change in the length of the anchor cable. The vertical excursion of clump-lifting also increases and it is equal to 8.3, 11.5 and 14.6 m respectively for the anchor lengths of 600, 800 and 1000 m.

9. CONCLUSIONS

1. The study of guy cable force versus excursion curves reveals that the nonlinear effect is important in the.case of horizontal excursion; it is found to be quite small in the case of vertical excursion. The stiffness of the mooring line is the largest around the neutral position and it decreases drastically after complete lifting of clump weights from the sea bed. -

2. The effect of axial rigidity ‘EA’ of cable on the mooring line response is significant but it decreases with the increased value of axial rigidity. Thus, in the - case where the value of ‘EA’ is quite large the cable can be taken as inextensible in the analysis of the - mooring line. The influence of guy cable ‘EA’ is - greater than that of anchor cable ‘EA’.

3. The stiff characteristics of the mooring line are enhanced with the increase in the unit weight of the clump weights.

4. The stiff behaviour range of the mooring line

increases significantly with the increase in the length of the anchor cable; the decrease in stiffness is relatively small.

Acknowledgemenrs-This research has been sponsored by the Department of Ocean Development, Government of India. The authors wish to express their appreciation to D.O.D., Govt of India for financial support.

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REFERENCES

R. L. Webster, An application of the finite element method to the determination of nonlinear static and dynamic responses of underwater cable structures. Ph.D. Dissertation, Cornell University, Ithaca, NY (1953). R. P. Singh, and J. Verma, Mooring line analysis for floating production platform. Interim Project Report, Department of Ocean Development, Government of India (1986). W. T. O’Brien and A. J. Francis, Cable movements under two-dimensional loading. J. Sfrucr. Div., ASCE 90, 89-123 (1964). W. T. O’Brien, General solution of suspended cable problems. J. Sfrucf. Div., ASCE 93, 1-26 (1967). A. Jennings, Discussion of cable movements under two-dimensional loading, by W. Terence O’Brien and Arthur J. Francis. J. Sfrucf. Diu., ASCE 91, 307-311 (1965). H. B. Jayaraman and W. C. Knudson, A curved element for the analysis of cable structures. Cornput. Sfrucf. 14, 325-333 (1981).